Properties

Degree 44
Conductor $ 3^{44} \cdot 23^{22} \cdot 29^{22} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 9·4-s − 6·7-s + 36·8-s − 28·13-s + 18·14-s + 26·16-s − 10·17-s − 8·19-s + 22·23-s − 55·25-s + 84·26-s + 54·28-s + 22·29-s − 18·31-s − 188·32-s + 30·34-s − 28·37-s + 24·38-s − 10·41-s − 14·43-s − 66·46-s − 18·47-s − 58·49-s + 165·50-s + 252·52-s + 20·53-s + ⋯
L(s)  = 1  − 2.12·2-s − 9/2·4-s − 2.26·7-s + 12.7·8-s − 7.76·13-s + 4.81·14-s + 13/2·16-s − 2.42·17-s − 1.83·19-s + 4.58·23-s − 11·25-s + 16.4·26-s + 10.2·28-s + 4.08·29-s − 3.23·31-s − 33.2·32-s + 5.14·34-s − 4.60·37-s + 3.89·38-s − 1.56·41-s − 2.13·43-s − 9.73·46-s − 2.62·47-s − 8.28·49-s + 23.3·50-s + 34.9·52-s + 2.74·53-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{44} \cdot 23^{22} \cdot 29^{22}\right)^{s/2} \, \Gamma_{\C}(s)^{22} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{44} \cdot 23^{22} \cdot 29^{22}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{22} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(44\)
\( N \)  =  \(3^{44} \cdot 23^{22} \cdot 29^{22}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6003} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  22
Selberg data  =  $(44,\ 3^{44} \cdot 23^{22} \cdot 29^{22} ,\ ( \ : [1/2]^{22} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;23,\;29\}$, \(F_p\) is a polynomial of degree 44. If $p \in \{3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 43.
$p$$F_p$
bad3 \( 1 \)
23 \( ( 1 - T )^{22} \)
29 \( ( 1 - T )^{22} \)
good2 \( 1 + 3 T + 9 p T^{2} + 45 T^{3} + 163 T^{4} + 89 p^{2} T^{5} + 1001 T^{6} + 1973 T^{7} + 2361 p T^{8} + 8583 T^{9} + 18313 T^{10} + 7791 p^{2} T^{11} + 60897 T^{12} + 98069 T^{13} + 178623 T^{14} + 137197 p T^{15} + 471693 T^{16} + 695305 T^{17} + 1138429 T^{18} + 202083 p^{3} T^{19} + 634473 p^{2} T^{20} + 869713 p^{2} T^{21} + 328693 p^{4} T^{22} + 869713 p^{3} T^{23} + 634473 p^{4} T^{24} + 202083 p^{6} T^{25} + 1138429 p^{4} T^{26} + 695305 p^{5} T^{27} + 471693 p^{6} T^{28} + 137197 p^{8} T^{29} + 178623 p^{8} T^{30} + 98069 p^{9} T^{31} + 60897 p^{10} T^{32} + 7791 p^{13} T^{33} + 18313 p^{12} T^{34} + 8583 p^{13} T^{35} + 2361 p^{15} T^{36} + 1973 p^{15} T^{37} + 1001 p^{16} T^{38} + 89 p^{19} T^{39} + 163 p^{18} T^{40} + 45 p^{19} T^{41} + 9 p^{21} T^{42} + 3 p^{21} T^{43} + p^{22} T^{44} \)
5 \( 1 + 11 p T^{2} - 8 T^{3} + 1519 T^{4} - 404 T^{5} + 27999 T^{6} - 10228 T^{7} + 77221 p T^{8} - 34606 p T^{9} + 4235824 T^{10} - 2197614 T^{11} + 38450676 T^{12} - 22322244 T^{13} + 59473883 p T^{14} - 7550346 p^{2} T^{15} + 2009378681 T^{16} - 1366593962 T^{17} + 12152671403 T^{18} - 345789462 p^{2} T^{19} + 67302413666 T^{20} - 48444249206 T^{21} + 347382399984 T^{22} - 48444249206 p T^{23} + 67302413666 p^{2} T^{24} - 345789462 p^{5} T^{25} + 12152671403 p^{4} T^{26} - 1366593962 p^{5} T^{27} + 2009378681 p^{6} T^{28} - 7550346 p^{9} T^{29} + 59473883 p^{9} T^{30} - 22322244 p^{9} T^{31} + 38450676 p^{10} T^{32} - 2197614 p^{11} T^{33} + 4235824 p^{12} T^{34} - 34606 p^{14} T^{35} + 77221 p^{15} T^{36} - 10228 p^{15} T^{37} + 27999 p^{16} T^{38} - 404 p^{17} T^{39} + 1519 p^{18} T^{40} - 8 p^{19} T^{41} + 11 p^{21} T^{42} + p^{22} T^{44} \)
7 \( 1 + 6 T + 94 T^{2} + 496 T^{3} + 4356 T^{4} + 20574 T^{5} + 133095 T^{6} + 570636 T^{7} + 3021644 T^{8} + 11889582 T^{9} + 54390232 T^{10} + 28298770 p T^{11} + 808021244 T^{12} + 2741846752 T^{13} + 1453569646 p T^{14} + 32329003306 T^{15} + 110624170063 T^{16} + 330291835838 T^{17} + 150248800938 p T^{18} + 2957446966188 T^{19} + 1259746321108 p T^{20} + 23373998495936 T^{21} + 65520051323302 T^{22} + 23373998495936 p T^{23} + 1259746321108 p^{3} T^{24} + 2957446966188 p^{3} T^{25} + 150248800938 p^{5} T^{26} + 330291835838 p^{5} T^{27} + 110624170063 p^{6} T^{28} + 32329003306 p^{7} T^{29} + 1453569646 p^{9} T^{30} + 2741846752 p^{9} T^{31} + 808021244 p^{10} T^{32} + 28298770 p^{12} T^{33} + 54390232 p^{12} T^{34} + 11889582 p^{13} T^{35} + 3021644 p^{14} T^{36} + 570636 p^{15} T^{37} + 133095 p^{16} T^{38} + 20574 p^{17} T^{39} + 4356 p^{18} T^{40} + 496 p^{19} T^{41} + 94 p^{20} T^{42} + 6 p^{21} T^{43} + p^{22} T^{44} \)
11 \( 1 + 133 T^{2} - 4 T^{3} + 8764 T^{4} - 544 T^{5} + 381310 T^{6} - 29692 T^{7} + 1119501 p T^{8} - 827528 T^{9} + 314742091 T^{10} - 8365336 T^{11} + 6633687246 T^{12} + 24963382 p T^{13} + 118787339157 T^{14} + 15198105344 T^{15} + 168291599304 p T^{16} + 403385161758 T^{17} + 25635789506004 T^{18} + 7459396049146 T^{19} + 321067171389254 T^{20} + 104891898181426 T^{21} + 3682231726987202 T^{22} + 104891898181426 p T^{23} + 321067171389254 p^{2} T^{24} + 7459396049146 p^{3} T^{25} + 25635789506004 p^{4} T^{26} + 403385161758 p^{5} T^{27} + 168291599304 p^{7} T^{28} + 15198105344 p^{7} T^{29} + 118787339157 p^{8} T^{30} + 24963382 p^{10} T^{31} + 6633687246 p^{10} T^{32} - 8365336 p^{11} T^{33} + 314742091 p^{12} T^{34} - 827528 p^{13} T^{35} + 1119501 p^{15} T^{36} - 29692 p^{15} T^{37} + 381310 p^{16} T^{38} - 544 p^{17} T^{39} + 8764 p^{18} T^{40} - 4 p^{19} T^{41} + 133 p^{20} T^{42} + p^{22} T^{44} \)
13 \( 1 + 28 T + 536 T^{2} + 7650 T^{3} + 91310 T^{4} + 938276 T^{5} + 8588221 T^{6} + 71046780 T^{7} + 539421459 T^{8} + 3790795080 T^{9} + 24867293062 T^{10} + 153091348260 T^{11} + 889118039870 T^{12} + 4888773818204 T^{13} + 25535824254995 T^{14} + 9770354127948 p T^{15} + 602973650046141 T^{16} + 2736400121312974 T^{17} + 11889010592819462 T^{18} + 49502648320043364 T^{19} + 197705714867836052 T^{20} + 757760821558888620 T^{21} + 2788335967694004946 T^{22} + 757760821558888620 p T^{23} + 197705714867836052 p^{2} T^{24} + 49502648320043364 p^{3} T^{25} + 11889010592819462 p^{4} T^{26} + 2736400121312974 p^{5} T^{27} + 602973650046141 p^{6} T^{28} + 9770354127948 p^{8} T^{29} + 25535824254995 p^{8} T^{30} + 4888773818204 p^{9} T^{31} + 889118039870 p^{10} T^{32} + 153091348260 p^{11} T^{33} + 24867293062 p^{12} T^{34} + 3790795080 p^{13} T^{35} + 539421459 p^{14} T^{36} + 71046780 p^{15} T^{37} + 8588221 p^{16} T^{38} + 938276 p^{17} T^{39} + 91310 p^{18} T^{40} + 7650 p^{19} T^{41} + 536 p^{20} T^{42} + 28 p^{21} T^{43} + p^{22} T^{44} \)
17 \( 1 + 10 T + 251 T^{2} + 122 p T^{3} + 29485 T^{4} + 210580 T^{5} + 2203111 T^{6} + 14023696 T^{7} + 119305232 T^{8} + 692307330 T^{9} + 5040802465 T^{10} + 27127205962 T^{11} + 174279340343 T^{12} + 51830247024 p T^{13} + 5096525624450 T^{14} + 24429615975990 T^{15} + 129112213551123 T^{16} + 590005269889510 T^{17} + 2883239123022128 T^{18} + 12584563220197970 T^{19} + 57450056154314560 T^{20} + 14066524742646958 p T^{21} + 1028894526561351718 T^{22} + 14066524742646958 p^{2} T^{23} + 57450056154314560 p^{2} T^{24} + 12584563220197970 p^{3} T^{25} + 2883239123022128 p^{4} T^{26} + 590005269889510 p^{5} T^{27} + 129112213551123 p^{6} T^{28} + 24429615975990 p^{7} T^{29} + 5096525624450 p^{8} T^{30} + 51830247024 p^{10} T^{31} + 174279340343 p^{10} T^{32} + 27127205962 p^{11} T^{33} + 5040802465 p^{12} T^{34} + 692307330 p^{13} T^{35} + 119305232 p^{14} T^{36} + 14023696 p^{15} T^{37} + 2203111 p^{16} T^{38} + 210580 p^{17} T^{39} + 29485 p^{18} T^{40} + 122 p^{20} T^{41} + 251 p^{20} T^{42} + 10 p^{21} T^{43} + p^{22} T^{44} \)
19 \( 1 + 8 T + 243 T^{2} + 1948 T^{3} + 30246 T^{4} + 234692 T^{5} + 2545597 T^{6} + 18666784 T^{7} + 446839 p^{2} T^{8} + 1103040616 T^{9} + 8137871748 T^{10} + 51633135176 T^{11} + 338247168453 T^{12} + 1992098997840 T^{13} + 11859201617127 T^{14} + 65034242963300 T^{15} + 356820129423882 T^{16} + 1828949737694652 T^{17} + 9331131910274201 T^{18} + 44849107683424984 T^{19} + 213984246059717899 T^{20} + 966282102382940608 T^{21} + 227695919080388072 p T^{22} + 966282102382940608 p T^{23} + 213984246059717899 p^{2} T^{24} + 44849107683424984 p^{3} T^{25} + 9331131910274201 p^{4} T^{26} + 1828949737694652 p^{5} T^{27} + 356820129423882 p^{6} T^{28} + 65034242963300 p^{7} T^{29} + 11859201617127 p^{8} T^{30} + 1992098997840 p^{9} T^{31} + 338247168453 p^{10} T^{32} + 51633135176 p^{11} T^{33} + 8137871748 p^{12} T^{34} + 1103040616 p^{13} T^{35} + 446839 p^{16} T^{36} + 18666784 p^{15} T^{37} + 2545597 p^{16} T^{38} + 234692 p^{17} T^{39} + 30246 p^{18} T^{40} + 1948 p^{19} T^{41} + 243 p^{20} T^{42} + 8 p^{21} T^{43} + p^{22} T^{44} \)
31 \( 1 + 18 T + 544 T^{2} + 7524 T^{3} + 135344 T^{4} + 1561710 T^{5} + 21364393 T^{6} + 214712490 T^{7} + 2444810586 T^{8} + 21953210942 T^{9} + 217646999070 T^{10} + 1774897153898 T^{11} + 15726442492061 T^{12} + 117744309588204 T^{13} + 947856982102181 T^{14} + 6563481961170304 T^{15} + 48528525398490318 T^{16} + 312313875995490678 T^{17} + 2136369543837164592 T^{18} + 12816778827416878456 T^{19} + 81502025627241029882 T^{20} + \)\(45\!\cdots\!16\)\( T^{21} + \)\(27\!\cdots\!24\)\( T^{22} + \)\(45\!\cdots\!16\)\( p T^{23} + 81502025627241029882 p^{2} T^{24} + 12816778827416878456 p^{3} T^{25} + 2136369543837164592 p^{4} T^{26} + 312313875995490678 p^{5} T^{27} + 48528525398490318 p^{6} T^{28} + 6563481961170304 p^{7} T^{29} + 947856982102181 p^{8} T^{30} + 117744309588204 p^{9} T^{31} + 15726442492061 p^{10} T^{32} + 1774897153898 p^{11} T^{33} + 217646999070 p^{12} T^{34} + 21953210942 p^{13} T^{35} + 2444810586 p^{14} T^{36} + 214712490 p^{15} T^{37} + 21364393 p^{16} T^{38} + 1561710 p^{17} T^{39} + 135344 p^{18} T^{40} + 7524 p^{19} T^{41} + 544 p^{20} T^{42} + 18 p^{21} T^{43} + p^{22} T^{44} \)
37 \( 1 + 28 T + 758 T^{2} + 13296 T^{3} + 219525 T^{4} + 2909880 T^{5} + 36523863 T^{6} + 10700092 p T^{7} + 4099823166 T^{8} + 37965467598 T^{9} + 339129577917 T^{10} + 2766054156288 T^{11} + 22013842136346 T^{12} + 162520079864552 T^{13} + 1187465661773071 T^{14} + 8161497339652380 T^{15} + 56357426707593093 T^{16} + 369982573376098480 T^{17} + 66632385097480324 p T^{18} + 15674407142221171580 T^{19} + \)\(10\!\cdots\!11\)\( T^{20} + \)\(62\!\cdots\!86\)\( T^{21} + \)\(38\!\cdots\!38\)\( T^{22} + \)\(62\!\cdots\!86\)\( p T^{23} + \)\(10\!\cdots\!11\)\( p^{2} T^{24} + 15674407142221171580 p^{3} T^{25} + 66632385097480324 p^{5} T^{26} + 369982573376098480 p^{5} T^{27} + 56357426707593093 p^{6} T^{28} + 8161497339652380 p^{7} T^{29} + 1187465661773071 p^{8} T^{30} + 162520079864552 p^{9} T^{31} + 22013842136346 p^{10} T^{32} + 2766054156288 p^{11} T^{33} + 339129577917 p^{12} T^{34} + 37965467598 p^{13} T^{35} + 4099823166 p^{14} T^{36} + 10700092 p^{16} T^{37} + 36523863 p^{16} T^{38} + 2909880 p^{17} T^{39} + 219525 p^{18} T^{40} + 13296 p^{19} T^{41} + 758 p^{20} T^{42} + 28 p^{21} T^{43} + p^{22} T^{44} \)
41 \( 1 + 10 T + 477 T^{2} + 4418 T^{3} + 112611 T^{4} + 960778 T^{5} + 17426925 T^{6} + 136378262 T^{7} + 1977185349 T^{8} + 14141633404 T^{9} + 174619040504 T^{10} + 1137287844324 T^{11} + 12471936166082 T^{12} + 73654318351716 T^{13} + 741665160434591 T^{14} + 3955625423238276 T^{15} + 37776699110610555 T^{16} + 181954736530660116 T^{17} + 1707260255103531813 T^{18} + 7523367748801243828 T^{19} + 71736761841597528050 T^{20} + \)\(30\!\cdots\!80\)\( T^{21} + \)\(29\!\cdots\!68\)\( T^{22} + \)\(30\!\cdots\!80\)\( p T^{23} + 71736761841597528050 p^{2} T^{24} + 7523367748801243828 p^{3} T^{25} + 1707260255103531813 p^{4} T^{26} + 181954736530660116 p^{5} T^{27} + 37776699110610555 p^{6} T^{28} + 3955625423238276 p^{7} T^{29} + 741665160434591 p^{8} T^{30} + 73654318351716 p^{9} T^{31} + 12471936166082 p^{10} T^{32} + 1137287844324 p^{11} T^{33} + 174619040504 p^{12} T^{34} + 14141633404 p^{13} T^{35} + 1977185349 p^{14} T^{36} + 136378262 p^{15} T^{37} + 17426925 p^{16} T^{38} + 960778 p^{17} T^{39} + 112611 p^{18} T^{40} + 4418 p^{19} T^{41} + 477 p^{20} T^{42} + 10 p^{21} T^{43} + p^{22} T^{44} \)
43 \( 1 + 14 T + 499 T^{2} + 6018 T^{3} + 119219 T^{4} + 1278592 T^{5} + 18462291 T^{6} + 178802384 T^{7} + 2094255063 T^{8} + 18450841206 T^{9} + 185491612926 T^{10} + 1493274303490 T^{11} + 13335878162402 T^{12} + 98528332719028 T^{13} + 800522451030051 T^{14} + 5462368821780322 T^{15} + 41235519380146953 T^{16} + 262775028279246726 T^{17} + 1889502219235857391 T^{18} + 11477256838369085614 T^{19} + 81091111375130980354 T^{20} + \)\(48\!\cdots\!14\)\( T^{21} + \)\(34\!\cdots\!68\)\( T^{22} + \)\(48\!\cdots\!14\)\( p T^{23} + 81091111375130980354 p^{2} T^{24} + 11477256838369085614 p^{3} T^{25} + 1889502219235857391 p^{4} T^{26} + 262775028279246726 p^{5} T^{27} + 41235519380146953 p^{6} T^{28} + 5462368821780322 p^{7} T^{29} + 800522451030051 p^{8} T^{30} + 98528332719028 p^{9} T^{31} + 13335878162402 p^{10} T^{32} + 1493274303490 p^{11} T^{33} + 185491612926 p^{12} T^{34} + 18450841206 p^{13} T^{35} + 2094255063 p^{14} T^{36} + 178802384 p^{15} T^{37} + 18462291 p^{16} T^{38} + 1278592 p^{17} T^{39} + 119219 p^{18} T^{40} + 6018 p^{19} T^{41} + 499 p^{20} T^{42} + 14 p^{21} T^{43} + p^{22} T^{44} \)
47 \( 1 + 18 T + 732 T^{2} + 240 p T^{3} + 256309 T^{4} + 3457770 T^{5} + 57368186 T^{6} + 688533872 T^{7} + 9242512482 T^{8} + 99891691740 T^{9} + 1144061561355 T^{10} + 11243614668542 T^{11} + 113505440102303 T^{12} + 1023148303530558 T^{13} + 9314803924787136 T^{14} + 77677034516258164 T^{15} + 649483462691246804 T^{16} + 5057410637107453634 T^{17} + 39454735289440565757 T^{18} + 6164746582613687910 p T^{19} + \)\(21\!\cdots\!13\)\( T^{20} + \)\(14\!\cdots\!08\)\( T^{21} + \)\(10\!\cdots\!32\)\( T^{22} + \)\(14\!\cdots\!08\)\( p T^{23} + \)\(21\!\cdots\!13\)\( p^{2} T^{24} + 6164746582613687910 p^{4} T^{25} + 39454735289440565757 p^{4} T^{26} + 5057410637107453634 p^{5} T^{27} + 649483462691246804 p^{6} T^{28} + 77677034516258164 p^{7} T^{29} + 9314803924787136 p^{8} T^{30} + 1023148303530558 p^{9} T^{31} + 113505440102303 p^{10} T^{32} + 11243614668542 p^{11} T^{33} + 1144061561355 p^{12} T^{34} + 99891691740 p^{13} T^{35} + 9242512482 p^{14} T^{36} + 688533872 p^{15} T^{37} + 57368186 p^{16} T^{38} + 3457770 p^{17} T^{39} + 256309 p^{18} T^{40} + 240 p^{20} T^{41} + 732 p^{20} T^{42} + 18 p^{21} T^{43} + p^{22} T^{44} \)
53 \( 1 - 20 T + 771 T^{2} - 11454 T^{3} + 263520 T^{4} - 3216654 T^{5} + 57198838 T^{6} - 607459354 T^{7} + 9191927315 T^{8} - 87743435192 T^{9} + 1181764085402 T^{10} - 10329304059232 T^{11} + 126859047248854 T^{12} - 1027243698831558 T^{13} + 11674853674587530 T^{14} - 88282653822495614 T^{15} + 937152520871909133 T^{16} - 6653279650650647576 T^{17} + 66352035404003691939 T^{18} - \)\(44\!\cdots\!98\)\( T^{19} + \)\(41\!\cdots\!69\)\( T^{20} - \)\(26\!\cdots\!80\)\( T^{21} + \)\(23\!\cdots\!40\)\( T^{22} - \)\(26\!\cdots\!80\)\( p T^{23} + \)\(41\!\cdots\!69\)\( p^{2} T^{24} - \)\(44\!\cdots\!98\)\( p^{3} T^{25} + 66352035404003691939 p^{4} T^{26} - 6653279650650647576 p^{5} T^{27} + 937152520871909133 p^{6} T^{28} - 88282653822495614 p^{7} T^{29} + 11674853674587530 p^{8} T^{30} - 1027243698831558 p^{9} T^{31} + 126859047248854 p^{10} T^{32} - 10329304059232 p^{11} T^{33} + 1181764085402 p^{12} T^{34} - 87743435192 p^{13} T^{35} + 9191927315 p^{14} T^{36} - 607459354 p^{15} T^{37} + 57198838 p^{16} T^{38} - 3216654 p^{17} T^{39} + 263520 p^{18} T^{40} - 11454 p^{19} T^{41} + 771 p^{20} T^{42} - 20 p^{21} T^{43} + p^{22} T^{44} \)
59 \( 1 + 20 T + 820 T^{2} + 12952 T^{3} + 304219 T^{4} + 3986962 T^{5} + 69192962 T^{6} + 771602070 T^{7} + 10879700362 T^{8} + 104173196090 T^{9} + 1251658832886 T^{10} + 10207331430010 T^{11} + 107462251201246 T^{12} + 716826917142692 T^{13} + 6738651312656588 T^{14} + 31596847218743628 T^{15} + 271086681108777805 T^{16} + 117230032138236046 T^{17} + 1687586043878732630 T^{18} - \)\(12\!\cdots\!74\)\( T^{19} - \)\(72\!\cdots\!21\)\( T^{20} - \)\(12\!\cdots\!76\)\( T^{21} - \)\(63\!\cdots\!12\)\( T^{22} - \)\(12\!\cdots\!76\)\( p T^{23} - \)\(72\!\cdots\!21\)\( p^{2} T^{24} - \)\(12\!\cdots\!74\)\( p^{3} T^{25} + 1687586043878732630 p^{4} T^{26} + 117230032138236046 p^{5} T^{27} + 271086681108777805 p^{6} T^{28} + 31596847218743628 p^{7} T^{29} + 6738651312656588 p^{8} T^{30} + 716826917142692 p^{9} T^{31} + 107462251201246 p^{10} T^{32} + 10207331430010 p^{11} T^{33} + 1251658832886 p^{12} T^{34} + 104173196090 p^{13} T^{35} + 10879700362 p^{14} T^{36} + 771602070 p^{15} T^{37} + 69192962 p^{16} T^{38} + 3986962 p^{17} T^{39} + 304219 p^{18} T^{40} + 12952 p^{19} T^{41} + 820 p^{20} T^{42} + 20 p^{21} T^{43} + p^{22} T^{44} \)
61 \( 1 + 38 T + 1387 T^{2} + 34928 T^{3} + 803164 T^{4} + 15632464 T^{5} + 280508888 T^{6} + 4538123788 T^{7} + 68577808213 T^{8} + 960869700572 T^{9} + 12707495942117 T^{10} + 158226656024842 T^{11} + 1874100298635415 T^{12} + 21098773092302830 T^{13} + 227254355925887593 T^{14} + 2341340453399799772 T^{15} + 23176039194974257928 T^{16} + \)\(22\!\cdots\!02\)\( T^{17} + \)\(20\!\cdots\!44\)\( T^{18} + \)\(17\!\cdots\!78\)\( T^{19} + \)\(15\!\cdots\!31\)\( T^{20} + \)\(12\!\cdots\!66\)\( T^{21} + \)\(99\!\cdots\!98\)\( T^{22} + \)\(12\!\cdots\!66\)\( p T^{23} + \)\(15\!\cdots\!31\)\( p^{2} T^{24} + \)\(17\!\cdots\!78\)\( p^{3} T^{25} + \)\(20\!\cdots\!44\)\( p^{4} T^{26} + \)\(22\!\cdots\!02\)\( p^{5} T^{27} + 23176039194974257928 p^{6} T^{28} + 2341340453399799772 p^{7} T^{29} + 227254355925887593 p^{8} T^{30} + 21098773092302830 p^{9} T^{31} + 1874100298635415 p^{10} T^{32} + 158226656024842 p^{11} T^{33} + 12707495942117 p^{12} T^{34} + 960869700572 p^{13} T^{35} + 68577808213 p^{14} T^{36} + 4538123788 p^{15} T^{37} + 280508888 p^{16} T^{38} + 15632464 p^{17} T^{39} + 803164 p^{18} T^{40} + 34928 p^{19} T^{41} + 1387 p^{20} T^{42} + 38 p^{21} T^{43} + p^{22} T^{44} \)
67 \( 1 + 50 T + 1815 T^{2} + 49282 T^{3} + 1133343 T^{4} + 22580442 T^{5} + 403683995 T^{6} + 6566408002 T^{7} + 98719470226 T^{8} + 1383905532342 T^{9} + 18250729171661 T^{10} + 3399521749024 p T^{11} + 2705062112286297 T^{12} + 30696694934613426 T^{13} + 334077499327270904 T^{14} + 3496462802930063058 T^{15} + 35277307743966867801 T^{16} + \)\(34\!\cdots\!56\)\( T^{17} + \)\(32\!\cdots\!46\)\( T^{18} + \)\(29\!\cdots\!90\)\( T^{19} + \)\(26\!\cdots\!64\)\( T^{20} + \)\(22\!\cdots\!08\)\( T^{21} + \)\(18\!\cdots\!62\)\( T^{22} + \)\(22\!\cdots\!08\)\( p T^{23} + \)\(26\!\cdots\!64\)\( p^{2} T^{24} + \)\(29\!\cdots\!90\)\( p^{3} T^{25} + \)\(32\!\cdots\!46\)\( p^{4} T^{26} + \)\(34\!\cdots\!56\)\( p^{5} T^{27} + 35277307743966867801 p^{6} T^{28} + 3496462802930063058 p^{7} T^{29} + 334077499327270904 p^{8} T^{30} + 30696694934613426 p^{9} T^{31} + 2705062112286297 p^{10} T^{32} + 3399521749024 p^{12} T^{33} + 18250729171661 p^{12} T^{34} + 1383905532342 p^{13} T^{35} + 98719470226 p^{14} T^{36} + 6566408002 p^{15} T^{37} + 403683995 p^{16} T^{38} + 22580442 p^{17} T^{39} + 1133343 p^{18} T^{40} + 49282 p^{19} T^{41} + 1815 p^{20} T^{42} + 50 p^{21} T^{43} + p^{22} T^{44} \)
71 \( 1 - 12 T + 893 T^{2} - 10608 T^{3} + 406273 T^{4} - 4672866 T^{5} + 124313123 T^{6} - 1367507960 T^{7} + 28568453791 T^{8} - 298668787786 T^{9} + 5231359863625 T^{10} - 51822097487834 T^{11} + 791906026606571 T^{12} - 7424348539281934 T^{13} + 101596212212532919 T^{14} - 901137417338351580 T^{15} + 11242914129724097871 T^{16} - 94327334997874094942 T^{17} + \)\(10\!\cdots\!77\)\( T^{18} - \)\(86\!\cdots\!14\)\( T^{19} + \)\(92\!\cdots\!69\)\( T^{20} - \)\(69\!\cdots\!84\)\( T^{21} + \)\(69\!\cdots\!58\)\( T^{22} - \)\(69\!\cdots\!84\)\( p T^{23} + \)\(92\!\cdots\!69\)\( p^{2} T^{24} - \)\(86\!\cdots\!14\)\( p^{3} T^{25} + \)\(10\!\cdots\!77\)\( p^{4} T^{26} - 94327334997874094942 p^{5} T^{27} + 11242914129724097871 p^{6} T^{28} - 901137417338351580 p^{7} T^{29} + 101596212212532919 p^{8} T^{30} - 7424348539281934 p^{9} T^{31} + 791906026606571 p^{10} T^{32} - 51822097487834 p^{11} T^{33} + 5231359863625 p^{12} T^{34} - 298668787786 p^{13} T^{35} + 28568453791 p^{14} T^{36} - 1367507960 p^{15} T^{37} + 124313123 p^{16} T^{38} - 4672866 p^{17} T^{39} + 406273 p^{18} T^{40} - 10608 p^{19} T^{41} + 893 p^{20} T^{42} - 12 p^{21} T^{43} + p^{22} T^{44} \)
73 \( 1 + 46 T + 1803 T^{2} + 50216 T^{3} + 1245492 T^{4} + 26335268 T^{5} + 510553489 T^{6} + 8951460330 T^{7} + 146510643852 T^{8} + 2228602075710 T^{9} + 32017335036030 T^{10} + 433965777757282 T^{11} + 5599430138516265 T^{12} + 68792553985499698 T^{13} + 808982565167303160 T^{14} + 9112299843469253554 T^{15} + 98623351753146979494 T^{16} + \)\(10\!\cdots\!36\)\( T^{17} + \)\(10\!\cdots\!35\)\( T^{18} + \)\(99\!\cdots\!26\)\( T^{19} + \)\(92\!\cdots\!64\)\( T^{20} + \)\(83\!\cdots\!46\)\( T^{21} + \)\(72\!\cdots\!74\)\( T^{22} + \)\(83\!\cdots\!46\)\( p T^{23} + \)\(92\!\cdots\!64\)\( p^{2} T^{24} + \)\(99\!\cdots\!26\)\( p^{3} T^{25} + \)\(10\!\cdots\!35\)\( p^{4} T^{26} + \)\(10\!\cdots\!36\)\( p^{5} T^{27} + 98623351753146979494 p^{6} T^{28} + 9112299843469253554 p^{7} T^{29} + 808982565167303160 p^{8} T^{30} + 68792553985499698 p^{9} T^{31} + 5599430138516265 p^{10} T^{32} + 433965777757282 p^{11} T^{33} + 32017335036030 p^{12} T^{34} + 2228602075710 p^{13} T^{35} + 146510643852 p^{14} T^{36} + 8951460330 p^{15} T^{37} + 510553489 p^{16} T^{38} + 26335268 p^{17} T^{39} + 1245492 p^{18} T^{40} + 50216 p^{19} T^{41} + 1803 p^{20} T^{42} + 46 p^{21} T^{43} + p^{22} T^{44} \)
79 \( 1 + 20 T + 904 T^{2} + 15626 T^{3} + 407340 T^{4} + 78146 p T^{5} + 121461873 T^{6} + 1647319212 T^{7} + 27038353914 T^{8} + 334005750154 T^{9} + 4809991847266 T^{10} + 54859826787824 T^{11} + 714073979000609 T^{12} + 7597953480837258 T^{13} + 91144683675595981 T^{14} + 912189433445988932 T^{15} + 10224155488412866664 T^{16} + 96883689949393175794 T^{17} + \)\(10\!\cdots\!74\)\( T^{18} + \)\(92\!\cdots\!42\)\( T^{19} + \)\(92\!\cdots\!08\)\( T^{20} + \)\(80\!\cdots\!48\)\( T^{21} + \)\(76\!\cdots\!16\)\( T^{22} + \)\(80\!\cdots\!48\)\( p T^{23} + \)\(92\!\cdots\!08\)\( p^{2} T^{24} + \)\(92\!\cdots\!42\)\( p^{3} T^{25} + \)\(10\!\cdots\!74\)\( p^{4} T^{26} + 96883689949393175794 p^{5} T^{27} + 10224155488412866664 p^{6} T^{28} + 912189433445988932 p^{7} T^{29} + 91144683675595981 p^{8} T^{30} + 7597953480837258 p^{9} T^{31} + 714073979000609 p^{10} T^{32} + 54859826787824 p^{11} T^{33} + 4809991847266 p^{12} T^{34} + 334005750154 p^{13} T^{35} + 27038353914 p^{14} T^{36} + 1647319212 p^{15} T^{37} + 121461873 p^{16} T^{38} + 78146 p^{18} T^{39} + 407340 p^{18} T^{40} + 15626 p^{19} T^{41} + 904 p^{20} T^{42} + 20 p^{21} T^{43} + p^{22} T^{44} \)
83 \( 1 - 22 T + 1337 T^{2} - 25632 T^{3} + 852422 T^{4} - 14396956 T^{5} + 345410024 T^{6} - 5193617170 T^{7} + 100140001429 T^{8} - 1353574984994 T^{9} + 22194827527046 T^{10} - 272137612041310 T^{11} + 3929609380091370 T^{12} - 44088836126281898 T^{13} + 574520598063981968 T^{14} - 5950912354514184506 T^{15} + 71325541424726841753 T^{16} - \)\(68\!\cdots\!72\)\( T^{17} + \)\(77\!\cdots\!41\)\( T^{18} - \)\(70\!\cdots\!22\)\( T^{19} + \)\(74\!\cdots\!49\)\( T^{20} - \)\(63\!\cdots\!22\)\( T^{21} + \)\(64\!\cdots\!76\)\( T^{22} - \)\(63\!\cdots\!22\)\( p T^{23} + \)\(74\!\cdots\!49\)\( p^{2} T^{24} - \)\(70\!\cdots\!22\)\( p^{3} T^{25} + \)\(77\!\cdots\!41\)\( p^{4} T^{26} - \)\(68\!\cdots\!72\)\( p^{5} T^{27} + 71325541424726841753 p^{6} T^{28} - 5950912354514184506 p^{7} T^{29} + 574520598063981968 p^{8} T^{30} - 44088836126281898 p^{9} T^{31} + 3929609380091370 p^{10} T^{32} - 272137612041310 p^{11} T^{33} + 22194827527046 p^{12} T^{34} - 1353574984994 p^{13} T^{35} + 100140001429 p^{14} T^{36} - 5193617170 p^{15} T^{37} + 345410024 p^{16} T^{38} - 14396956 p^{17} T^{39} + 852422 p^{18} T^{40} - 25632 p^{19} T^{41} + 1337 p^{20} T^{42} - 22 p^{21} T^{43} + p^{22} T^{44} \)
89 \( 1 + 14 T + 894 T^{2} + 9346 T^{3} + 362695 T^{4} + 2853344 T^{5} + 91557533 T^{6} + 539831502 T^{7} + 16718512470 T^{8} + 76275379344 T^{9} + 2441722918363 T^{10} + 9950067041920 T^{11} + 303807662299937 T^{12} + 1399097419082682 T^{13} + 32961480291575422 T^{14} + 203548155929738262 T^{15} + 3131161780614694857 T^{16} + 27599918492386868418 T^{17} + \)\(26\!\cdots\!71\)\( T^{18} + \)\(33\!\cdots\!66\)\( T^{19} + \)\(21\!\cdots\!48\)\( T^{20} + \)\(34\!\cdots\!38\)\( T^{21} + \)\(18\!\cdots\!74\)\( T^{22} + \)\(34\!\cdots\!38\)\( p T^{23} + \)\(21\!\cdots\!48\)\( p^{2} T^{24} + \)\(33\!\cdots\!66\)\( p^{3} T^{25} + \)\(26\!\cdots\!71\)\( p^{4} T^{26} + 27599918492386868418 p^{5} T^{27} + 3131161780614694857 p^{6} T^{28} + 203548155929738262 p^{7} T^{29} + 32961480291575422 p^{8} T^{30} + 1399097419082682 p^{9} T^{31} + 303807662299937 p^{10} T^{32} + 9950067041920 p^{11} T^{33} + 2441722918363 p^{12} T^{34} + 76275379344 p^{13} T^{35} + 16718512470 p^{14} T^{36} + 539831502 p^{15} T^{37} + 91557533 p^{16} T^{38} + 2853344 p^{17} T^{39} + 362695 p^{18} T^{40} + 9346 p^{19} T^{41} + 894 p^{20} T^{42} + 14 p^{21} T^{43} + p^{22} T^{44} \)
97 \( 1 + 48 T + 2401 T^{2} + 74808 T^{3} + 2248455 T^{4} + 53150176 T^{5} + 1200368159 T^{6} + 23095640426 T^{7} + 425211126570 T^{8} + 6919480578648 T^{9} + 108162169027616 T^{10} + 1522849461127426 T^{11} + 20686920417223960 T^{12} + 255463519354128292 T^{13} + 3060314488808522878 T^{14} + 33438435948817360600 T^{15} + \)\(35\!\cdots\!59\)\( T^{16} + \)\(34\!\cdots\!14\)\( T^{17} + \)\(34\!\cdots\!43\)\( T^{18} + \)\(30\!\cdots\!88\)\( T^{19} + \)\(28\!\cdots\!07\)\( T^{20} + \)\(25\!\cdots\!06\)\( T^{21} + \)\(25\!\cdots\!42\)\( T^{22} + \)\(25\!\cdots\!06\)\( p T^{23} + \)\(28\!\cdots\!07\)\( p^{2} T^{24} + \)\(30\!\cdots\!88\)\( p^{3} T^{25} + \)\(34\!\cdots\!43\)\( p^{4} T^{26} + \)\(34\!\cdots\!14\)\( p^{5} T^{27} + \)\(35\!\cdots\!59\)\( p^{6} T^{28} + 33438435948817360600 p^{7} T^{29} + 3060314488808522878 p^{8} T^{30} + 255463519354128292 p^{9} T^{31} + 20686920417223960 p^{10} T^{32} + 1522849461127426 p^{11} T^{33} + 108162169027616 p^{12} T^{34} + 6919480578648 p^{13} T^{35} + 425211126570 p^{14} T^{36} + 23095640426 p^{15} T^{37} + 1200368159 p^{16} T^{38} + 53150176 p^{17} T^{39} + 2248455 p^{18} T^{40} + 74808 p^{19} T^{41} + 2401 p^{20} T^{42} + 48 p^{21} T^{43} + p^{22} T^{44} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{44} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.97177777547200688962353415067, −1.86516315426915866092447823202, −1.80842188250684125179402193733, −1.80791011172606773274090474886, −1.76398297602110708434762341913, −1.75227669797089786275276216124, −1.73740181225608367248466492218, −1.71303345871422803056652458446, −1.47064425424072349201002076702, −1.42651356827798640288312651825, −1.35700629050175259863854802370, −1.34287562312540155761585199622, −1.30720716999374041212564945384, −1.28433317694643548021924972750, −1.26149258139602070093908203531, −1.25913244539314970333823178733, −1.24424332575653202718190008361, −1.15258697518118553789476517391, −1.14561483121253766794721599419, −1.09869798412030860395334704737, −1.09416265554881086901775926429, −0.998680315493724787531339416015, −0.866377066549047050069742269284, −0.857962473595032970441260868976, −0.821068179556457909540779569515, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.821068179556457909540779569515, 0.857962473595032970441260868976, 0.866377066549047050069742269284, 0.998680315493724787531339416015, 1.09416265554881086901775926429, 1.09869798412030860395334704737, 1.14561483121253766794721599419, 1.15258697518118553789476517391, 1.24424332575653202718190008361, 1.25913244539314970333823178733, 1.26149258139602070093908203531, 1.28433317694643548021924972750, 1.30720716999374041212564945384, 1.34287562312540155761585199622, 1.35700629050175259863854802370, 1.42651356827798640288312651825, 1.47064425424072349201002076702, 1.71303345871422803056652458446, 1.73740181225608367248466492218, 1.75227669797089786275276216124, 1.76398297602110708434762341913, 1.80791011172606773274090474886, 1.80842188250684125179402193733, 1.86516315426915866092447823202, 1.97177777547200688962353415067

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

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