Properties

Degree 46
Conductor $ 2^{69} \cdot 17^{23} \cdot 59^{23} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 23

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s − 7-s − 5·9-s − 3·11-s − 7·13-s − 23·17-s − 16·19-s + 6·21-s − 29·23-s − 42·25-s + 107·27-s − 5·29-s − 41·31-s + 18·33-s + 5·37-s + 42·39-s + 11·41-s + 13·43-s − 39·47-s − 72·49-s + 138·51-s − 2·53-s + 96·57-s − 23·59-s − 37·61-s + 5·63-s − 34·67-s + ⋯
L(s)  = 1  − 3.46·3-s − 0.377·7-s − 5/3·9-s − 0.904·11-s − 1.94·13-s − 5.57·17-s − 3.67·19-s + 1.30·21-s − 6.04·23-s − 8.39·25-s + 20.5·27-s − 0.928·29-s − 7.36·31-s + 3.13·33-s + 0.821·37-s + 6.72·39-s + 1.71·41-s + 1.98·43-s − 5.68·47-s − 10.2·49-s + 19.3·51-s − 0.274·53-s + 12.7·57-s − 2.99·59-s − 4.73·61-s + 0.629·63-s − 4.15·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{69} \cdot 17^{23} \cdot 59^{23}\right)^{s/2} \, \Gamma_{\C}(s)^{23} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{69} \cdot 17^{23} \cdot 59^{23}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{23} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(46\)
\( N \)  =  \(2^{69} \cdot 17^{23} \cdot 59^{23}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8024} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  23
Selberg data  =  $(46,\ 2^{69} \cdot 17^{23} \cdot 59^{23} ,\ ( \ : [1/2]^{23} ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \(F_p\) is a polynomial of degree 46. If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 45.
$p$$F_p$
bad2 \( 1 \)
17 \( ( 1 + T )^{23} \)
59 \( ( 1 + T )^{23} \)
good3 \( 1 + 2 p T + 41 T^{2} + 169 T^{3} + 80 p^{2} T^{4} + 799 p T^{5} + 8053 T^{6} + 23248 T^{7} + 67136 T^{8} + 174523 T^{9} + 452066 T^{10} + 1081715 T^{11} + 2574451 T^{12} + 1916467 p T^{13} + 12762458 T^{14} + 26853011 T^{15} + 56155606 T^{16} + 112055411 T^{17} + 8232053 p^{3} T^{18} + 422498612 T^{19} + 88716859 p^{2} T^{20} + 1449832918 T^{21} + 2617687424 T^{22} + 1515583070 p T^{23} + 2617687424 p T^{24} + 1449832918 p^{2} T^{25} + 88716859 p^{5} T^{26} + 422498612 p^{4} T^{27} + 8232053 p^{8} T^{28} + 112055411 p^{6} T^{29} + 56155606 p^{7} T^{30} + 26853011 p^{8} T^{31} + 12762458 p^{9} T^{32} + 1916467 p^{11} T^{33} + 2574451 p^{11} T^{34} + 1081715 p^{12} T^{35} + 452066 p^{13} T^{36} + 174523 p^{14} T^{37} + 67136 p^{15} T^{38} + 23248 p^{16} T^{39} + 8053 p^{17} T^{40} + 799 p^{19} T^{41} + 80 p^{21} T^{42} + 169 p^{20} T^{43} + 41 p^{21} T^{44} + 2 p^{23} T^{45} + p^{23} T^{46} \)
5 \( 1 + 42 T^{2} - 28 T^{3} + 934 T^{4} - 1119 T^{5} + 14986 T^{6} - 23617 T^{7} + 38818 p T^{8} - 352168 T^{9} + 2130738 T^{10} - 4148472 T^{11} + 20335249 T^{12} - 40782198 T^{13} + 34269957 p T^{14} - 344972721 T^{15} + 1288030546 T^{16} - 2558241481 T^{17} + 8697703172 T^{18} - 16830643802 T^{19} + 53006956087 T^{20} - 98956650418 T^{21} + 292396648558 T^{22} - 522027462432 T^{23} + 292396648558 p T^{24} - 98956650418 p^{2} T^{25} + 53006956087 p^{3} T^{26} - 16830643802 p^{4} T^{27} + 8697703172 p^{5} T^{28} - 2558241481 p^{6} T^{29} + 1288030546 p^{7} T^{30} - 344972721 p^{8} T^{31} + 34269957 p^{10} T^{32} - 40782198 p^{10} T^{33} + 20335249 p^{11} T^{34} - 4148472 p^{12} T^{35} + 2130738 p^{13} T^{36} - 352168 p^{14} T^{37} + 38818 p^{16} T^{38} - 23617 p^{16} T^{39} + 14986 p^{17} T^{40} - 1119 p^{18} T^{41} + 934 p^{19} T^{42} - 28 p^{20} T^{43} + 42 p^{21} T^{44} + p^{23} T^{46} \)
7 \( 1 + T + 73 T^{2} + 79 T^{3} + 2743 T^{4} + 3026 T^{5} + 70241 T^{6} + 75655 T^{7} + 1372095 T^{8} + 1393603 T^{9} + 21726750 T^{10} + 20199306 T^{11} + 289759606 T^{12} + 240235655 T^{13} + 3343118803 T^{14} + 2419089195 T^{15} + 34051578067 T^{16} + 21224208150 T^{17} + 311121177257 T^{18} + 167450017655 T^{19} + 2581877178397 T^{20} + 1231411401725 T^{21} + 19632773615201 T^{22} + 8722094954908 T^{23} + 19632773615201 p T^{24} + 1231411401725 p^{2} T^{25} + 2581877178397 p^{3} T^{26} + 167450017655 p^{4} T^{27} + 311121177257 p^{5} T^{28} + 21224208150 p^{6} T^{29} + 34051578067 p^{7} T^{30} + 2419089195 p^{8} T^{31} + 3343118803 p^{9} T^{32} + 240235655 p^{10} T^{33} + 289759606 p^{11} T^{34} + 20199306 p^{12} T^{35} + 21726750 p^{13} T^{36} + 1393603 p^{14} T^{37} + 1372095 p^{15} T^{38} + 75655 p^{16} T^{39} + 70241 p^{17} T^{40} + 3026 p^{18} T^{41} + 2743 p^{19} T^{42} + 79 p^{20} T^{43} + 73 p^{21} T^{44} + p^{22} T^{45} + p^{23} T^{46} \)
11 \( 1 + 3 T + 139 T^{2} + 302 T^{3} + 9314 T^{4} + 13367 T^{5} + 407136 T^{6} + 308265 T^{7} + 13253905 T^{8} + 1852876 T^{9} + 347196643 T^{10} - 137246479 T^{11} + 698869267 p T^{12} - 6320577258 T^{13} + 148246561338 T^{14} - 167202132525 T^{15} + 2530540325164 T^{16} - 3337383083280 T^{17} + 38537150988455 T^{18} - 54474687936313 T^{19} + 525589502841786 T^{20} - 752133261792564 T^{21} + 6433864202354528 T^{22} - 8912608778648100 T^{23} + 6433864202354528 p T^{24} - 752133261792564 p^{2} T^{25} + 525589502841786 p^{3} T^{26} - 54474687936313 p^{4} T^{27} + 38537150988455 p^{5} T^{28} - 3337383083280 p^{6} T^{29} + 2530540325164 p^{7} T^{30} - 167202132525 p^{8} T^{31} + 148246561338 p^{9} T^{32} - 6320577258 p^{10} T^{33} + 698869267 p^{12} T^{34} - 137246479 p^{12} T^{35} + 347196643 p^{13} T^{36} + 1852876 p^{14} T^{37} + 13253905 p^{15} T^{38} + 308265 p^{16} T^{39} + 407136 p^{17} T^{40} + 13367 p^{18} T^{41} + 9314 p^{19} T^{42} + 302 p^{20} T^{43} + 139 p^{21} T^{44} + 3 p^{22} T^{45} + p^{23} T^{46} \)
13 \( 1 + 7 T + 189 T^{2} + 1237 T^{3} + 17863 T^{4} + 108387 T^{5} + 1116536 T^{6} + 482370 p T^{7} + 51670300 T^{8} + 269101220 T^{9} + 1883042622 T^{10} + 9124103939 T^{11} + 4323211435 p T^{12} + 254306284859 T^{13} + 1411535747339 T^{14} + 5986038909756 T^{15} + 30425468017112 T^{16} + 121302715246588 T^{17} + 571051528641558 T^{18} + 165026958735024 p T^{19} + 9429922605248869 T^{20} + 33425057122243827 T^{21} + 137933441877062508 T^{22} + 461279945817760804 T^{23} + 137933441877062508 p T^{24} + 33425057122243827 p^{2} T^{25} + 9429922605248869 p^{3} T^{26} + 165026958735024 p^{5} T^{27} + 571051528641558 p^{5} T^{28} + 121302715246588 p^{6} T^{29} + 30425468017112 p^{7} T^{30} + 5986038909756 p^{8} T^{31} + 1411535747339 p^{9} T^{32} + 254306284859 p^{10} T^{33} + 4323211435 p^{12} T^{34} + 9124103939 p^{12} T^{35} + 1883042622 p^{13} T^{36} + 269101220 p^{14} T^{37} + 51670300 p^{15} T^{38} + 482370 p^{17} T^{39} + 1116536 p^{17} T^{40} + 108387 p^{18} T^{41} + 17863 p^{19} T^{42} + 1237 p^{20} T^{43} + 189 p^{21} T^{44} + 7 p^{22} T^{45} + p^{23} T^{46} \)
19 \( 1 + 16 T + 303 T^{2} + 3575 T^{3} + 41936 T^{4} + 399829 T^{5} + 3656732 T^{6} + 29676810 T^{7} + 229793921 T^{8} + 1641011236 T^{9} + 11214016712 T^{10} + 72116448586 T^{11} + 446063787283 T^{12} + 2628266297021 T^{13} + 14972968940470 T^{14} + 81924403082054 T^{15} + 435410466308605 T^{16} + 2235635852973770 T^{17} + 11191504465137057 T^{18} + 54352051708270115 T^{19} + 258025367992283716 T^{20} + 1191628250822826957 T^{21} + 5387637850277118472 T^{22} + 23728936823042854294 T^{23} + 5387637850277118472 p T^{24} + 1191628250822826957 p^{2} T^{25} + 258025367992283716 p^{3} T^{26} + 54352051708270115 p^{4} T^{27} + 11191504465137057 p^{5} T^{28} + 2235635852973770 p^{6} T^{29} + 435410466308605 p^{7} T^{30} + 81924403082054 p^{8} T^{31} + 14972968940470 p^{9} T^{32} + 2628266297021 p^{10} T^{33} + 446063787283 p^{11} T^{34} + 72116448586 p^{12} T^{35} + 11214016712 p^{13} T^{36} + 1641011236 p^{14} T^{37} + 229793921 p^{15} T^{38} + 29676810 p^{16} T^{39} + 3656732 p^{17} T^{40} + 399829 p^{18} T^{41} + 41936 p^{19} T^{42} + 3575 p^{20} T^{43} + 303 p^{21} T^{44} + 16 p^{22} T^{45} + p^{23} T^{46} \)
23 \( 1 + 29 T + 682 T^{2} + 11499 T^{3} + 169187 T^{4} + 2121542 T^{5} + 24128949 T^{6} + 247643597 T^{7} + 2358268963 T^{8} + 20829193805 T^{9} + 173195311220 T^{10} + 1356864954813 T^{11} + 10105113524264 T^{12} + 71608081493726 T^{13} + 485591664789218 T^{14} + 3153545126037781 T^{15} + 19687682789668587 T^{16} + 118213898988531457 T^{17} + 684462754224629500 T^{18} + 3822284686722196724 T^{19} + 20623769412134153183 T^{20} + \)\(10\!\cdots\!49\)\( T^{21} + \)\(54\!\cdots\!04\)\( T^{22} + \)\(26\!\cdots\!24\)\( T^{23} + \)\(54\!\cdots\!04\)\( p T^{24} + \)\(10\!\cdots\!49\)\( p^{2} T^{25} + 20623769412134153183 p^{3} T^{26} + 3822284686722196724 p^{4} T^{27} + 684462754224629500 p^{5} T^{28} + 118213898988531457 p^{6} T^{29} + 19687682789668587 p^{7} T^{30} + 3153545126037781 p^{8} T^{31} + 485591664789218 p^{9} T^{32} + 71608081493726 p^{10} T^{33} + 10105113524264 p^{11} T^{34} + 1356864954813 p^{12} T^{35} + 173195311220 p^{13} T^{36} + 20829193805 p^{14} T^{37} + 2358268963 p^{15} T^{38} + 247643597 p^{16} T^{39} + 24128949 p^{17} T^{40} + 2121542 p^{18} T^{41} + 169187 p^{19} T^{42} + 11499 p^{20} T^{43} + 682 p^{21} T^{44} + 29 p^{22} T^{45} + p^{23} T^{46} \)
29 \( 1 + 5 T + 361 T^{2} + 1455 T^{3} + 63271 T^{4} + 209299 T^{5} + 7260623 T^{6} + 20082503 T^{7} + 618239543 T^{8} + 1467935086 T^{9} + 1444373316 p T^{10} + 88653532050 T^{11} + 2361516164436 T^{12} + 4664936151186 T^{13} + 114299654360024 T^{14} + 220228320822959 T^{15} + 4859839626052520 T^{16} + 9411526638250669 T^{17} + 184744840663098908 T^{18} + 362826258473712075 T^{19} + 6364886183768286284 T^{20} + 12528038749916340475 T^{21} + \)\(20\!\cdots\!37\)\( T^{22} + \)\(38\!\cdots\!44\)\( T^{23} + \)\(20\!\cdots\!37\)\( p T^{24} + 12528038749916340475 p^{2} T^{25} + 6364886183768286284 p^{3} T^{26} + 362826258473712075 p^{4} T^{27} + 184744840663098908 p^{5} T^{28} + 9411526638250669 p^{6} T^{29} + 4859839626052520 p^{7} T^{30} + 220228320822959 p^{8} T^{31} + 114299654360024 p^{9} T^{32} + 4664936151186 p^{10} T^{33} + 2361516164436 p^{11} T^{34} + 88653532050 p^{12} T^{35} + 1444373316 p^{14} T^{36} + 1467935086 p^{14} T^{37} + 618239543 p^{15} T^{38} + 20082503 p^{16} T^{39} + 7260623 p^{17} T^{40} + 209299 p^{18} T^{41} + 63271 p^{19} T^{42} + 1455 p^{20} T^{43} + 361 p^{21} T^{44} + 5 p^{22} T^{45} + p^{23} T^{46} \)
31 \( 1 + 41 T + 1249 T^{2} + 27755 T^{3} + 524372 T^{4} + 8461333 T^{5} + 122223707 T^{6} + 1589505781 T^{7} + 19008812540 T^{8} + 6773892270 p T^{9} + 2167960254185 T^{10} + 20987646328417 T^{11} + 191895765804949 T^{12} + 53585216756966 p T^{13} + 13679526997533490 T^{14} + 107348636253586234 T^{15} + 805436910161859297 T^{16} + 5784493494724588025 T^{17} + 39858668738280468086 T^{18} + \)\(26\!\cdots\!72\)\( T^{19} + \)\(16\!\cdots\!87\)\( T^{20} + \)\(10\!\cdots\!65\)\( T^{21} + \)\(60\!\cdots\!81\)\( T^{22} + \)\(34\!\cdots\!34\)\( T^{23} + \)\(60\!\cdots\!81\)\( p T^{24} + \)\(10\!\cdots\!65\)\( p^{2} T^{25} + \)\(16\!\cdots\!87\)\( p^{3} T^{26} + \)\(26\!\cdots\!72\)\( p^{4} T^{27} + 39858668738280468086 p^{5} T^{28} + 5784493494724588025 p^{6} T^{29} + 805436910161859297 p^{7} T^{30} + 107348636253586234 p^{8} T^{31} + 13679526997533490 p^{9} T^{32} + 53585216756966 p^{11} T^{33} + 191895765804949 p^{11} T^{34} + 20987646328417 p^{12} T^{35} + 2167960254185 p^{13} T^{36} + 6773892270 p^{15} T^{37} + 19008812540 p^{15} T^{38} + 1589505781 p^{16} T^{39} + 122223707 p^{17} T^{40} + 8461333 p^{18} T^{41} + 524372 p^{19} T^{42} + 27755 p^{20} T^{43} + 1249 p^{21} T^{44} + 41 p^{22} T^{45} + p^{23} T^{46} \)
37 \( 1 - 5 T + 515 T^{2} - 2079 T^{3} + 128825 T^{4} - 409718 T^{5} + 20993729 T^{6} - 50449783 T^{7} + 2521137752 T^{8} - 4275643006 T^{9} + 239204687297 T^{10} - 254489757112 T^{11} + 18762470891685 T^{12} - 9776625654024 T^{13} + 1255437112769261 T^{14} - 105017165129542 T^{15} + 73239496162416128 T^{16} + 17416499304889383 T^{17} + 3778848751857172188 T^{18} + 1676006628620981023 T^{19} + \)\(17\!\cdots\!06\)\( T^{20} + 94600870565241923706 T^{21} + \)\(71\!\cdots\!53\)\( T^{22} + \)\(39\!\cdots\!74\)\( T^{23} + \)\(71\!\cdots\!53\)\( p T^{24} + 94600870565241923706 p^{2} T^{25} + \)\(17\!\cdots\!06\)\( p^{3} T^{26} + 1676006628620981023 p^{4} T^{27} + 3778848751857172188 p^{5} T^{28} + 17416499304889383 p^{6} T^{29} + 73239496162416128 p^{7} T^{30} - 105017165129542 p^{8} T^{31} + 1255437112769261 p^{9} T^{32} - 9776625654024 p^{10} T^{33} + 18762470891685 p^{11} T^{34} - 254489757112 p^{12} T^{35} + 239204687297 p^{13} T^{36} - 4275643006 p^{14} T^{37} + 2521137752 p^{15} T^{38} - 50449783 p^{16} T^{39} + 20993729 p^{17} T^{40} - 409718 p^{18} T^{41} + 128825 p^{19} T^{42} - 2079 p^{20} T^{43} + 515 p^{21} T^{44} - 5 p^{22} T^{45} + p^{23} T^{46} \)
41 \( 1 - 11 T + 610 T^{2} - 5779 T^{3} + 178075 T^{4} - 1470058 T^{5} + 33335152 T^{6} - 241446283 T^{7} + 4524175264 T^{8} - 28868723252 T^{9} + 477712642354 T^{10} - 2694401560063 T^{11} + 41184385728456 T^{12} - 206269747620891 T^{13} + 3005859592659244 T^{14} - 13472840698661812 T^{15} + 190893753923309562 T^{16} - 774682085089907354 T^{17} + 10751238526130998035 T^{18} - 40067472021782973298 T^{19} + \)\(54\!\cdots\!26\)\( T^{20} - \)\(18\!\cdots\!90\)\( T^{21} + \)\(24\!\cdots\!89\)\( T^{22} - \)\(19\!\cdots\!90\)\( p T^{23} + \)\(24\!\cdots\!89\)\( p T^{24} - \)\(18\!\cdots\!90\)\( p^{2} T^{25} + \)\(54\!\cdots\!26\)\( p^{3} T^{26} - 40067472021782973298 p^{4} T^{27} + 10751238526130998035 p^{5} T^{28} - 774682085089907354 p^{6} T^{29} + 190893753923309562 p^{7} T^{30} - 13472840698661812 p^{8} T^{31} + 3005859592659244 p^{9} T^{32} - 206269747620891 p^{10} T^{33} + 41184385728456 p^{11} T^{34} - 2694401560063 p^{12} T^{35} + 477712642354 p^{13} T^{36} - 28868723252 p^{14} T^{37} + 4524175264 p^{15} T^{38} - 241446283 p^{16} T^{39} + 33335152 p^{17} T^{40} - 1470058 p^{18} T^{41} + 178075 p^{19} T^{42} - 5779 p^{20} T^{43} + 610 p^{21} T^{44} - 11 p^{22} T^{45} + p^{23} T^{46} \)
43 \( 1 - 13 T + 515 T^{2} - 5315 T^{3} + 125138 T^{4} - 24961 p T^{5} + 19627843 T^{6} - 143397822 T^{7} + 2274861735 T^{8} - 14360732687 T^{9} + 210755782203 T^{10} - 1160589385926 T^{11} + 16444844830307 T^{12} - 79638738294666 T^{13} + 1120247112144483 T^{14} - 4814848917847800 T^{15} + 68239425045907190 T^{16} - 263411917646838171 T^{17} + 3773215608605446246 T^{18} - 13284935255760489639 T^{19} + \)\(19\!\cdots\!31\)\( T^{20} - \)\(62\!\cdots\!76\)\( T^{21} + \)\(89\!\cdots\!76\)\( T^{22} - \)\(27\!\cdots\!00\)\( T^{23} + \)\(89\!\cdots\!76\)\( p T^{24} - \)\(62\!\cdots\!76\)\( p^{2} T^{25} + \)\(19\!\cdots\!31\)\( p^{3} T^{26} - 13284935255760489639 p^{4} T^{27} + 3773215608605446246 p^{5} T^{28} - 263411917646838171 p^{6} T^{29} + 68239425045907190 p^{7} T^{30} - 4814848917847800 p^{8} T^{31} + 1120247112144483 p^{9} T^{32} - 79638738294666 p^{10} T^{33} + 16444844830307 p^{11} T^{34} - 1160589385926 p^{12} T^{35} + 210755782203 p^{13} T^{36} - 14360732687 p^{14} T^{37} + 2274861735 p^{15} T^{38} - 143397822 p^{16} T^{39} + 19627843 p^{17} T^{40} - 24961 p^{19} T^{41} + 125138 p^{19} T^{42} - 5315 p^{20} T^{43} + 515 p^{21} T^{44} - 13 p^{22} T^{45} + p^{23} T^{46} \)
47 \( 1 + 39 T + 1241 T^{2} + 28457 T^{3} + 565576 T^{4} + 9580017 T^{5} + 145788522 T^{6} + 1992146681 T^{7} + 24969132703 T^{8} + 287623742825 T^{9} + 3076255155651 T^{10} + 30598973832977 T^{11} + 284502631458814 T^{12} + 2473946232208524 T^{13} + 20157463503980607 T^{14} + 153733185733770156 T^{15} + 1096460578201625604 T^{16} + 7290398799282646359 T^{17} + 45029130693320165013 T^{18} + \)\(25\!\cdots\!54\)\( T^{19} + \)\(13\!\cdots\!55\)\( T^{20} + \)\(67\!\cdots\!80\)\( T^{21} + \)\(34\!\cdots\!87\)\( T^{22} + \)\(20\!\cdots\!06\)\( T^{23} + \)\(34\!\cdots\!87\)\( p T^{24} + \)\(67\!\cdots\!80\)\( p^{2} T^{25} + \)\(13\!\cdots\!55\)\( p^{3} T^{26} + \)\(25\!\cdots\!54\)\( p^{4} T^{27} + 45029130693320165013 p^{5} T^{28} + 7290398799282646359 p^{6} T^{29} + 1096460578201625604 p^{7} T^{30} + 153733185733770156 p^{8} T^{31} + 20157463503980607 p^{9} T^{32} + 2473946232208524 p^{10} T^{33} + 284502631458814 p^{11} T^{34} + 30598973832977 p^{12} T^{35} + 3076255155651 p^{13} T^{36} + 287623742825 p^{14} T^{37} + 24969132703 p^{15} T^{38} + 1992146681 p^{16} T^{39} + 145788522 p^{17} T^{40} + 9580017 p^{18} T^{41} + 565576 p^{19} T^{42} + 28457 p^{20} T^{43} + 1241 p^{21} T^{44} + 39 p^{22} T^{45} + p^{23} T^{46} \)
53 \( 1 + 2 T + 757 T^{2} + 180 T^{3} + 276929 T^{4} - 406274 T^{5} + 66043920 T^{6} - 203759474 T^{7} + 220449787 p T^{8} - 52985272454 T^{9} + 1650491594783 T^{10} - 9425263896254 T^{11} + 194831842770047 T^{12} - 1272275169792773 T^{13} + 19749663225416491 T^{14} - 137751817512766068 T^{15} + 1744882081641679884 T^{16} - 12377439966604840595 T^{17} + \)\(13\!\cdots\!15\)\( T^{18} - \)\(94\!\cdots\!08\)\( T^{19} + \)\(92\!\cdots\!89\)\( T^{20} - \)\(61\!\cdots\!77\)\( T^{21} + \)\(55\!\cdots\!93\)\( T^{22} - \)\(35\!\cdots\!22\)\( T^{23} + \)\(55\!\cdots\!93\)\( p T^{24} - \)\(61\!\cdots\!77\)\( p^{2} T^{25} + \)\(92\!\cdots\!89\)\( p^{3} T^{26} - \)\(94\!\cdots\!08\)\( p^{4} T^{27} + \)\(13\!\cdots\!15\)\( p^{5} T^{28} - 12377439966604840595 p^{6} T^{29} + 1744882081641679884 p^{7} T^{30} - 137751817512766068 p^{8} T^{31} + 19749663225416491 p^{9} T^{32} - 1272275169792773 p^{10} T^{33} + 194831842770047 p^{11} T^{34} - 9425263896254 p^{12} T^{35} + 1650491594783 p^{13} T^{36} - 52985272454 p^{14} T^{37} + 220449787 p^{16} T^{38} - 203759474 p^{16} T^{39} + 66043920 p^{17} T^{40} - 406274 p^{18} T^{41} + 276929 p^{19} T^{42} + 180 p^{20} T^{43} + 757 p^{21} T^{44} + 2 p^{22} T^{45} + p^{23} T^{46} \)
61 \( 1 + 37 T + 1497 T^{2} + 37554 T^{3} + 930721 T^{4} + 18286202 T^{5} + 348427543 T^{6} + 5736961009 T^{7} + 91346834704 T^{8} + 1309700123709 T^{9} + 18177357046559 T^{10} + 232465485090409 T^{11} + 2883257921965852 T^{12} + 33420430706964540 T^{13} + 376397085940096594 T^{14} + 3998085183890315398 T^{15} + 41328163882432862806 T^{16} + \)\(40\!\cdots\!58\)\( T^{17} + \)\(38\!\cdots\!78\)\( T^{18} + \)\(35\!\cdots\!87\)\( T^{19} + \)\(31\!\cdots\!61\)\( T^{20} + \)\(26\!\cdots\!98\)\( T^{21} + \)\(21\!\cdots\!36\)\( T^{22} + \)\(17\!\cdots\!62\)\( T^{23} + \)\(21\!\cdots\!36\)\( p T^{24} + \)\(26\!\cdots\!98\)\( p^{2} T^{25} + \)\(31\!\cdots\!61\)\( p^{3} T^{26} + \)\(35\!\cdots\!87\)\( p^{4} T^{27} + \)\(38\!\cdots\!78\)\( p^{5} T^{28} + \)\(40\!\cdots\!58\)\( p^{6} T^{29} + 41328163882432862806 p^{7} T^{30} + 3998085183890315398 p^{8} T^{31} + 376397085940096594 p^{9} T^{32} + 33420430706964540 p^{10} T^{33} + 2883257921965852 p^{11} T^{34} + 232465485090409 p^{12} T^{35} + 18177357046559 p^{13} T^{36} + 1309700123709 p^{14} T^{37} + 91346834704 p^{15} T^{38} + 5736961009 p^{16} T^{39} + 348427543 p^{17} T^{40} + 18286202 p^{18} T^{41} + 930721 p^{19} T^{42} + 37554 p^{20} T^{43} + 1497 p^{21} T^{44} + 37 p^{22} T^{45} + p^{23} T^{46} \)
67 \( 1 + 34 T + 1450 T^{2} + 33398 T^{3} + 836157 T^{4} + 14693024 T^{5} + 272089156 T^{6} + 3847100585 T^{7} + 57334116924 T^{8} + 668151713650 T^{9} + 124971758297 p T^{10} + 80783134959909 T^{11} + 879154572695732 T^{12} + 6908055809126424 T^{13} + 68184292792704936 T^{14} + 415688739201046294 T^{15} + 4127540225926867017 T^{16} + 17745477088958512306 T^{17} + \)\(23\!\cdots\!50\)\( T^{18} + \)\(66\!\cdots\!61\)\( T^{19} + \)\(15\!\cdots\!34\)\( T^{20} + \)\(39\!\cdots\!70\)\( T^{21} + \)\(11\!\cdots\!98\)\( T^{22} + \)\(29\!\cdots\!38\)\( T^{23} + \)\(11\!\cdots\!98\)\( p T^{24} + \)\(39\!\cdots\!70\)\( p^{2} T^{25} + \)\(15\!\cdots\!34\)\( p^{3} T^{26} + \)\(66\!\cdots\!61\)\( p^{4} T^{27} + \)\(23\!\cdots\!50\)\( p^{5} T^{28} + 17745477088958512306 p^{6} T^{29} + 4127540225926867017 p^{7} T^{30} + 415688739201046294 p^{8} T^{31} + 68184292792704936 p^{9} T^{32} + 6908055809126424 p^{10} T^{33} + 879154572695732 p^{11} T^{34} + 80783134959909 p^{12} T^{35} + 124971758297 p^{14} T^{36} + 668151713650 p^{14} T^{37} + 57334116924 p^{15} T^{38} + 3847100585 p^{16} T^{39} + 272089156 p^{17} T^{40} + 14693024 p^{18} T^{41} + 836157 p^{19} T^{42} + 33398 p^{20} T^{43} + 1450 p^{21} T^{44} + 34 p^{22} T^{45} + p^{23} T^{46} \)
71 \( 1 + 13 T + 951 T^{2} + 12075 T^{3} + 445824 T^{4} + 5469799 T^{5} + 137361911 T^{6} + 1617515358 T^{7} + 31309861908 T^{8} + 352812324759 T^{9} + 5640494171637 T^{10} + 60816362285911 T^{11} + 838694200725277 T^{12} + 8665723881484227 T^{13} + 106177876526312936 T^{14} + 1053121731079794208 T^{15} + 11709426171868091483 T^{16} + \)\(11\!\cdots\!00\)\( T^{17} + \)\(11\!\cdots\!76\)\( T^{18} + \)\(10\!\cdots\!10\)\( T^{19} + \)\(99\!\cdots\!39\)\( T^{20} + \)\(87\!\cdots\!78\)\( T^{21} + \)\(78\!\cdots\!05\)\( T^{22} + \)\(65\!\cdots\!32\)\( T^{23} + \)\(78\!\cdots\!05\)\( p T^{24} + \)\(87\!\cdots\!78\)\( p^{2} T^{25} + \)\(99\!\cdots\!39\)\( p^{3} T^{26} + \)\(10\!\cdots\!10\)\( p^{4} T^{27} + \)\(11\!\cdots\!76\)\( p^{5} T^{28} + \)\(11\!\cdots\!00\)\( p^{6} T^{29} + 11709426171868091483 p^{7} T^{30} + 1053121731079794208 p^{8} T^{31} + 106177876526312936 p^{9} T^{32} + 8665723881484227 p^{10} T^{33} + 838694200725277 p^{11} T^{34} + 60816362285911 p^{12} T^{35} + 5640494171637 p^{13} T^{36} + 352812324759 p^{14} T^{37} + 31309861908 p^{15} T^{38} + 1617515358 p^{16} T^{39} + 137361911 p^{17} T^{40} + 5469799 p^{18} T^{41} + 445824 p^{19} T^{42} + 12075 p^{20} T^{43} + 951 p^{21} T^{44} + 13 p^{22} T^{45} + p^{23} T^{46} \)
73 \( 1 + 14 T + 811 T^{2} + 10200 T^{3} + 337868 T^{4} + 3864267 T^{5} + 95607122 T^{6} + 1005218952 T^{7} + 20590583491 T^{8} + 200721051291 T^{9} + 3589917528870 T^{10} + 32665635280073 T^{11} + 526547014877961 T^{12} + 4496478550708604 T^{13} + 66675281050089039 T^{14} + 536713490160152287 T^{15} + 7422657559283624595 T^{16} + 56526647773765651493 T^{17} + 10081317744960101979 p T^{18} + \)\(53\!\cdots\!17\)\( T^{19} + \)\(65\!\cdots\!87\)\( T^{20} + \)\(45\!\cdots\!07\)\( T^{21} + \)\(52\!\cdots\!36\)\( T^{22} + \)\(34\!\cdots\!86\)\( T^{23} + \)\(52\!\cdots\!36\)\( p T^{24} + \)\(45\!\cdots\!07\)\( p^{2} T^{25} + \)\(65\!\cdots\!87\)\( p^{3} T^{26} + \)\(53\!\cdots\!17\)\( p^{4} T^{27} + 10081317744960101979 p^{6} T^{28} + 56526647773765651493 p^{6} T^{29} + 7422657559283624595 p^{7} T^{30} + 536713490160152287 p^{8} T^{31} + 66675281050089039 p^{9} T^{32} + 4496478550708604 p^{10} T^{33} + 526547014877961 p^{11} T^{34} + 32665635280073 p^{12} T^{35} + 3589917528870 p^{13} T^{36} + 200721051291 p^{14} T^{37} + 20590583491 p^{15} T^{38} + 1005218952 p^{16} T^{39} + 95607122 p^{17} T^{40} + 3864267 p^{18} T^{41} + 337868 p^{19} T^{42} + 10200 p^{20} T^{43} + 811 p^{21} T^{44} + 14 p^{22} T^{45} + p^{23} T^{46} \)
79 \( 1 + 61 T + 2696 T^{2} + 85181 T^{3} + 2258676 T^{4} + 50173223 T^{5} + 985808231 T^{6} + 17116686919 T^{7} + 270049030914 T^{8} + 48937722346 p T^{9} + 51170496226891 T^{10} + 625293003193171 T^{11} + 7171525590935889 T^{12} + 77189141453499800 T^{13} + 794592686434984730 T^{14} + 7843549594569009700 T^{15} + 76005520440836456743 T^{16} + \)\(72\!\cdots\!67\)\( T^{17} + \)\(69\!\cdots\!64\)\( T^{18} + \)\(66\!\cdots\!22\)\( T^{19} + \)\(63\!\cdots\!60\)\( T^{20} + \)\(59\!\cdots\!55\)\( T^{21} + \)\(55\!\cdots\!91\)\( T^{22} + \)\(49\!\cdots\!38\)\( T^{23} + \)\(55\!\cdots\!91\)\( p T^{24} + \)\(59\!\cdots\!55\)\( p^{2} T^{25} + \)\(63\!\cdots\!60\)\( p^{3} T^{26} + \)\(66\!\cdots\!22\)\( p^{4} T^{27} + \)\(69\!\cdots\!64\)\( p^{5} T^{28} + \)\(72\!\cdots\!67\)\( p^{6} T^{29} + 76005520440836456743 p^{7} T^{30} + 7843549594569009700 p^{8} T^{31} + 794592686434984730 p^{9} T^{32} + 77189141453499800 p^{10} T^{33} + 7171525590935889 p^{11} T^{34} + 625293003193171 p^{12} T^{35} + 51170496226891 p^{13} T^{36} + 48937722346 p^{15} T^{37} + 270049030914 p^{15} T^{38} + 17116686919 p^{16} T^{39} + 985808231 p^{17} T^{40} + 50173223 p^{18} T^{41} + 2258676 p^{19} T^{42} + 85181 p^{20} T^{43} + 2696 p^{21} T^{44} + 61 p^{22} T^{45} + p^{23} T^{46} \)
83 \( 1 + 9 T + 1050 T^{2} + 9435 T^{3} + 541934 T^{4} + 4923556 T^{5} + 184867576 T^{6} + 1706892544 T^{7} + 47218171719 T^{8} + 442053856741 T^{9} + 9678630110352 T^{10} + 91145077822239 T^{11} + 1662178888446734 T^{12} + 15572195439289142 T^{13} + 245901302877288557 T^{14} + 2265492101795048938 T^{15} + 31904684453259688130 T^{16} + \)\(28\!\cdots\!69\)\( T^{17} + \)\(36\!\cdots\!24\)\( T^{18} + \)\(31\!\cdots\!82\)\( T^{19} + \)\(37\!\cdots\!82\)\( T^{20} + \)\(31\!\cdots\!23\)\( T^{21} + \)\(34\!\cdots\!01\)\( T^{22} + \)\(27\!\cdots\!48\)\( T^{23} + \)\(34\!\cdots\!01\)\( p T^{24} + \)\(31\!\cdots\!23\)\( p^{2} T^{25} + \)\(37\!\cdots\!82\)\( p^{3} T^{26} + \)\(31\!\cdots\!82\)\( p^{4} T^{27} + \)\(36\!\cdots\!24\)\( p^{5} T^{28} + \)\(28\!\cdots\!69\)\( p^{6} T^{29} + 31904684453259688130 p^{7} T^{30} + 2265492101795048938 p^{8} T^{31} + 245901302877288557 p^{9} T^{32} + 15572195439289142 p^{10} T^{33} + 1662178888446734 p^{11} T^{34} + 91145077822239 p^{12} T^{35} + 9678630110352 p^{13} T^{36} + 442053856741 p^{14} T^{37} + 47218171719 p^{15} T^{38} + 1706892544 p^{16} T^{39} + 184867576 p^{17} T^{40} + 4923556 p^{18} T^{41} + 541934 p^{19} T^{42} + 9435 p^{20} T^{43} + 1050 p^{21} T^{44} + 9 p^{22} T^{45} + p^{23} T^{46} \)
89 \( 1 - 28 T + 1319 T^{2} - 27706 T^{3} + 777582 T^{4} - 13558041 T^{5} + 292094148 T^{6} - 4468970705 T^{7} + 81221578606 T^{8} - 1125064678956 T^{9} + 18056741703529 T^{10} - 230587051251980 T^{11} + 3353552361929952 T^{12} - 39934171637192770 T^{13} + 534946590816834724 T^{14} - 5985027032930619202 T^{15} + 74651664005516122473 T^{16} - \)\(78\!\cdots\!52\)\( T^{17} + \)\(92\!\cdots\!01\)\( T^{18} - \)\(10\!\cdots\!66\)\( p T^{19} + \)\(10\!\cdots\!37\)\( T^{20} - \)\(96\!\cdots\!21\)\( T^{21} + \)\(10\!\cdots\!00\)\( T^{22} - \)\(90\!\cdots\!90\)\( T^{23} + \)\(10\!\cdots\!00\)\( p T^{24} - \)\(96\!\cdots\!21\)\( p^{2} T^{25} + \)\(10\!\cdots\!37\)\( p^{3} T^{26} - \)\(10\!\cdots\!66\)\( p^{5} T^{27} + \)\(92\!\cdots\!01\)\( p^{5} T^{28} - \)\(78\!\cdots\!52\)\( p^{6} T^{29} + 74651664005516122473 p^{7} T^{30} - 5985027032930619202 p^{8} T^{31} + 534946590816834724 p^{9} T^{32} - 39934171637192770 p^{10} T^{33} + 3353552361929952 p^{11} T^{34} - 230587051251980 p^{12} T^{35} + 18056741703529 p^{13} T^{36} - 1125064678956 p^{14} T^{37} + 81221578606 p^{15} T^{38} - 4468970705 p^{16} T^{39} + 292094148 p^{17} T^{40} - 13558041 p^{18} T^{41} + 777582 p^{19} T^{42} - 27706 p^{20} T^{43} + 1319 p^{21} T^{44} - 28 p^{22} T^{45} + p^{23} T^{46} \)
97 \( 1 + 1011 T^{2} - 812 T^{3} + 511016 T^{4} - 831520 T^{5} + 173105855 T^{6} - 426128625 T^{7} + 44413408665 T^{8} - 145900965659 T^{9} + 9244264857570 T^{10} - 37537259157759 T^{11} + 1632439404486055 T^{12} - 7733305144418312 T^{13} + 252379637998950389 T^{14} - 1327882398990293351 T^{15} + 34919850279553823937 T^{16} - \)\(19\!\cdots\!69\)\( T^{17} + \)\(43\!\cdots\!85\)\( T^{18} - \)\(25\!\cdots\!99\)\( T^{19} + \)\(50\!\cdots\!71\)\( T^{20} - \)\(28\!\cdots\!36\)\( T^{21} + \)\(53\!\cdots\!93\)\( T^{22} - \)\(29\!\cdots\!04\)\( T^{23} + \)\(53\!\cdots\!93\)\( p T^{24} - \)\(28\!\cdots\!36\)\( p^{2} T^{25} + \)\(50\!\cdots\!71\)\( p^{3} T^{26} - \)\(25\!\cdots\!99\)\( p^{4} T^{27} + \)\(43\!\cdots\!85\)\( p^{5} T^{28} - \)\(19\!\cdots\!69\)\( p^{6} T^{29} + 34919850279553823937 p^{7} T^{30} - 1327882398990293351 p^{8} T^{31} + 252379637998950389 p^{9} T^{32} - 7733305144418312 p^{10} T^{33} + 1632439404486055 p^{11} T^{34} - 37537259157759 p^{12} T^{35} + 9244264857570 p^{13} T^{36} - 145900965659 p^{14} T^{37} + 44413408665 p^{15} T^{38} - 426128625 p^{16} T^{39} + 173105855 p^{17} T^{40} - 831520 p^{18} T^{41} + 511016 p^{19} T^{42} - 812 p^{20} T^{43} + 1011 p^{21} T^{44} + p^{23} T^{46} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{46} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.88334977096772249294810658707, −1.75173586141286990161274596569, −1.71526279348593872247722260416, −1.70316375004063620955654181229, −1.69834128185127170096220065441, −1.61544877776204764142287168069, −1.59554641514916828341744469039, −1.58041565350037373744932295601, −1.55335398052620210537217118422, −1.48772203313641968810649845750, −1.46519553442032322211272110548, −1.44540929860504133809559183375, −1.33947776525169476071038213668, −1.26100012301555654582753442500, −1.20490125433407660414199199464, −1.19452328392631576136237346938, −1.15808945461239984875275686049, −1.09358943032154410070140605365, −1.04307449331364630874134043064, −1.02263307443737199392330282849, −0.969366534397503258965905729998, −0.947294511350439989500358879279, −0.818790509958938628049838663111, −0.75239854780050374802032284045, −0.57394683759065298038374972875, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.57394683759065298038374972875, 0.75239854780050374802032284045, 0.818790509958938628049838663111, 0.947294511350439989500358879279, 0.969366534397503258965905729998, 1.02263307443737199392330282849, 1.04307449331364630874134043064, 1.09358943032154410070140605365, 1.15808945461239984875275686049, 1.19452328392631576136237346938, 1.20490125433407660414199199464, 1.26100012301555654582753442500, 1.33947776525169476071038213668, 1.44540929860504133809559183375, 1.46519553442032322211272110548, 1.48772203313641968810649845750, 1.55335398052620210537217118422, 1.58041565350037373744932295601, 1.59554641514916828341744469039, 1.61544877776204764142287168069, 1.69834128185127170096220065441, 1.70316375004063620955654181229, 1.71526279348593872247722260416, 1.75173586141286990161274596569, 1.88334977096772249294810658707

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.