Properties

Degree 64
Conductor $ 2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·2-s + 512·4-s − 4·7-s + 5.44e3·8-s + 8·9-s − 128·14-s + 4.29e4·16-s + 256·18-s − 24·23-s + 8·25-s − 2.04e3·28-s + 112·29-s + 2.67e5·32-s + 4.09e3·36-s + 32·37-s − 32·43-s − 768·46-s + 8·49-s + 256·50-s + 136·53-s − 2.17e4·56-s + 3.58e3·58-s − 32·63-s + 1.35e6·64-s + 32·67-s + 4.35e4·72-s + 1.02e3·74-s + ⋯
L(s)  = 1  + 16·2-s + 128·4-s − 4/7·7-s + 680·8-s + 8/9·9-s − 9.14·14-s + 2.68e3·16-s + 14.2·18-s − 1.04·23-s + 8/25·25-s − 73.1·28-s + 3.86·29-s + 8.36e3·32-s + 113.·36-s + 0.864·37-s − 0.744·43-s − 16.6·46-s + 8/49·49-s + 5.11·50-s + 2.56·53-s − 388.·56-s + 61.7·58-s − 0.507·63-s + 2.12e4·64-s + 0.477·67-s + 604.·72-s + 13.8·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{32} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{32} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(64\)
\( N \)  =  \(2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{210} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((64,\ 2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32} ,\ ( \ : [1]^{32} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(21.1651\)
\(L(\frac12)\)  \(\approx\)  \(21.1651\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 64. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 63.
$p$$F_p(T)$
bad2 \( ( 1 - p T + p T^{2} )^{16} \)
3 \( 1 - 8 T^{2} + 13 T^{4} - 736 T^{6} - 4198 T^{8} + 83728 T^{10} + 56099 T^{12} + 200 p^{5} T^{14} - 55186 p^{5} T^{16} + 200 p^{9} T^{18} + 56099 p^{8} T^{20} + 83728 p^{12} T^{22} - 4198 p^{16} T^{24} - 736 p^{20} T^{26} + 13 p^{24} T^{28} - 8 p^{28} T^{30} + p^{32} T^{32} \)
5 \( 1 - 8 T^{2} + 352 p T^{4} + 488 T^{6} + 1662396 T^{8} + 13332088 T^{10} + 55235616 p^{2} T^{12} + 16776488 p^{4} T^{14} + 12668894 p^{7} T^{16} + 16776488 p^{8} T^{18} + 55235616 p^{10} T^{20} + 13332088 p^{12} T^{22} + 1662396 p^{16} T^{24} + 488 p^{20} T^{26} + 352 p^{25} T^{28} - 8 p^{28} T^{30} + p^{32} T^{32} \)
7 \( 1 + 4 T + 8 T^{2} - 444 T^{3} + 2712 T^{4} + 2300 T^{5} + 12296 p T^{6} - 132508 p T^{7} + 18396092 T^{8} - 3719764 p T^{9} - 1434072 p^{2} T^{10} - 13393620 p^{3} T^{11} + 1339218472 p^{2} T^{12} - 131640428 p^{3} T^{13} + 129792280 p^{4} T^{14} - 620205548 p^{5} T^{15} + 1203339430 p^{6} T^{16} - 620205548 p^{7} T^{17} + 129792280 p^{8} T^{18} - 131640428 p^{9} T^{19} + 1339218472 p^{10} T^{20} - 13393620 p^{13} T^{21} - 1434072 p^{14} T^{22} - 3719764 p^{15} T^{23} + 18396092 p^{16} T^{24} - 132508 p^{19} T^{25} + 12296 p^{21} T^{26} + 2300 p^{22} T^{27} + 2712 p^{24} T^{28} - 444 p^{26} T^{29} + 8 p^{28} T^{30} + 4 p^{30} T^{31} + p^{32} T^{32} \)
good11 \( ( 1 - 854 T^{2} + 364021 T^{4} - 105164338 T^{6} + 23502428062 T^{8} - 4363116439174 T^{10} + 700309877045307 T^{12} - 99487346205684658 T^{14} + 12679799467137910946 T^{16} - 99487346205684658 p^{4} T^{18} + 700309877045307 p^{8} T^{20} - 4363116439174 p^{12} T^{22} + 23502428062 p^{16} T^{24} - 105164338 p^{20} T^{26} + 364021 p^{24} T^{28} - 854 p^{28} T^{30} + p^{32} T^{32} )^{2} \)
13 \( 1 - 41318 T^{4} + 1848534637 T^{8} - 21529314661158 T^{12} + 2043143406069299558 T^{16} - \)\(67\!\cdots\!38\)\( T^{20} + \)\(34\!\cdots\!95\)\( T^{24} - \)\(47\!\cdots\!90\)\( T^{28} + \)\(18\!\cdots\!86\)\( T^{32} - \)\(47\!\cdots\!90\)\( p^{8} T^{36} + \)\(34\!\cdots\!95\)\( p^{16} T^{40} - \)\(67\!\cdots\!38\)\( p^{24} T^{44} + 2043143406069299558 p^{32} T^{48} - 21529314661158 p^{40} T^{52} + 1848534637 p^{48} T^{56} - 41318 p^{56} T^{60} + p^{64} T^{64} \)
17 \( 1 - 529510 T^{4} + 136379745453 T^{8} - 23558491761084454 T^{12} + \)\(32\!\cdots\!54\)\( T^{16} - \)\(38\!\cdots\!10\)\( T^{20} + \)\(40\!\cdots\!19\)\( T^{24} - \)\(37\!\cdots\!02\)\( T^{28} + \)\(32\!\cdots\!78\)\( T^{32} - \)\(37\!\cdots\!02\)\( p^{8} T^{36} + \)\(40\!\cdots\!19\)\( p^{16} T^{40} - \)\(38\!\cdots\!10\)\( p^{24} T^{44} + \)\(32\!\cdots\!54\)\( p^{32} T^{48} - 23558491761084454 p^{40} T^{52} + 136379745453 p^{48} T^{56} - 529510 p^{56} T^{60} + p^{64} T^{64} \)
19 \( ( 1 + 2424 T^{2} + 3312224 T^{4} + 168860568 p T^{6} + 2429535700668 T^{8} + 79381483632840 p T^{10} + 789663727700940960 T^{12} + \)\(35\!\cdots\!48\)\( T^{14} + \)\(13\!\cdots\!46\)\( T^{16} + \)\(35\!\cdots\!48\)\( p^{4} T^{18} + 789663727700940960 p^{8} T^{20} + 79381483632840 p^{13} T^{22} + 2429535700668 p^{16} T^{24} + 168860568 p^{21} T^{26} + 3312224 p^{24} T^{28} + 2424 p^{28} T^{30} + p^{32} T^{32} )^{2} \)
23 \( ( 1 + 12 T + 72 T^{2} - 5252 T^{3} + 175028 T^{4} + 9417748 T^{5} + 114202712 T^{6} + 1648475588 T^{7} - 5478039556 p T^{8} + 140235357292 T^{9} + 26454980423880 T^{10} + 1754839525121628 T^{11} - 8341122994074804 T^{12} - 1125285162229151596 T^{13} - 9681243056979155816 T^{14} - \)\(10\!\cdots\!40\)\( T^{15} + \)\(12\!\cdots\!70\)\( T^{16} - \)\(10\!\cdots\!40\)\( p^{2} T^{17} - 9681243056979155816 p^{4} T^{18} - 1125285162229151596 p^{6} T^{19} - 8341122994074804 p^{8} T^{20} + 1754839525121628 p^{10} T^{21} + 26454980423880 p^{12} T^{22} + 140235357292 p^{14} T^{23} - 5478039556 p^{17} T^{24} + 1648475588 p^{18} T^{25} + 114202712 p^{20} T^{26} + 9417748 p^{22} T^{27} + 175028 p^{24} T^{28} - 5252 p^{26} T^{29} + 72 p^{28} T^{30} + 12 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
29 \( ( 1 - 28 T + 3901 T^{2} - 102040 T^{3} + 7908434 T^{4} - 186434464 T^{5} + 10617161811 T^{6} - 222510300940 T^{7} + 10319166884890 T^{8} - 222510300940 p^{2} T^{9} + 10617161811 p^{4} T^{10} - 186434464 p^{6} T^{11} + 7908434 p^{8} T^{12} - 102040 p^{10} T^{13} + 3901 p^{12} T^{14} - 28 p^{14} T^{15} + p^{16} T^{16} )^{4} \)
31 \( ( 1 - 7576 T^{2} + 30435296 T^{4} - 85129876392 T^{6} + 183038542019900 T^{8} - 317981825299831160 T^{10} + \)\(45\!\cdots\!80\)\( T^{12} - \)\(56\!\cdots\!16\)\( T^{14} + \)\(58\!\cdots\!14\)\( T^{16} - \)\(56\!\cdots\!16\)\( p^{4} T^{18} + \)\(45\!\cdots\!80\)\( p^{8} T^{20} - 317981825299831160 p^{12} T^{22} + 183038542019900 p^{16} T^{24} - 85129876392 p^{20} T^{26} + 30435296 p^{24} T^{28} - 7576 p^{28} T^{30} + p^{32} T^{32} )^{2} \)
37 \( ( 1 - 16 T + 128 T^{2} + 32256 T^{3} - 34548 p T^{4} - 1079936 p T^{5} + 1323166208 T^{6} - 11355718704 T^{7} - 7039170093180 T^{8} + 29591999553008 T^{9} + 2550929501783808 T^{10} - 391642728377707008 T^{11} + 3687624649437089988 T^{12} + \)\(53\!\cdots\!32\)\( T^{13} - \)\(10\!\cdots\!24\)\( T^{14} + \)\(83\!\cdots\!48\)\( T^{15} + \)\(28\!\cdots\!54\)\( T^{16} + \)\(83\!\cdots\!48\)\( p^{2} T^{17} - \)\(10\!\cdots\!24\)\( p^{4} T^{18} + \)\(53\!\cdots\!32\)\( p^{6} T^{19} + 3687624649437089988 p^{8} T^{20} - 391642728377707008 p^{10} T^{21} + 2550929501783808 p^{12} T^{22} + 29591999553008 p^{14} T^{23} - 7039170093180 p^{16} T^{24} - 11355718704 p^{18} T^{25} + 1323166208 p^{20} T^{26} - 1079936 p^{23} T^{27} - 34548 p^{25} T^{28} + 32256 p^{26} T^{29} + 128 p^{28} T^{30} - 16 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
41 \( ( 1 + 14120 T^{2} + 96143456 T^{4} + 424264619064 T^{6} + 1378935974050620 T^{8} + 3568204679473072616 T^{10} + \)\(77\!\cdots\!08\)\( T^{12} + \)\(14\!\cdots\!04\)\( T^{14} + \)\(26\!\cdots\!02\)\( T^{16} + \)\(14\!\cdots\!04\)\( p^{4} T^{18} + \)\(77\!\cdots\!08\)\( p^{8} T^{20} + 3568204679473072616 p^{12} T^{22} + 1378935974050620 p^{16} T^{24} + 424264619064 p^{20} T^{26} + 96143456 p^{24} T^{28} + 14120 p^{28} T^{30} + p^{32} T^{32} )^{2} \)
43 \( ( 1 + 16 T + 128 T^{2} - 27816 T^{3} - 7996188 T^{4} - 56141240 T^{5} + 512117152 T^{6} + 161747182896 T^{7} + 18366380121412 T^{8} - 18441194387632 T^{9} - 1474210965392608 T^{10} - 143352700380422120 T^{11} - 39977541767912872292 T^{12} - \)\(48\!\cdots\!36\)\( T^{13} - \)\(56\!\cdots\!20\)\( T^{14} + \)\(77\!\cdots\!76\)\( T^{15} + \)\(21\!\cdots\!30\)\( T^{16} + \)\(77\!\cdots\!76\)\( p^{2} T^{17} - \)\(56\!\cdots\!20\)\( p^{4} T^{18} - \)\(48\!\cdots\!36\)\( p^{6} T^{19} - 39977541767912872292 p^{8} T^{20} - 143352700380422120 p^{10} T^{21} - 1474210965392608 p^{12} T^{22} - 18441194387632 p^{14} T^{23} + 18366380121412 p^{16} T^{24} + 161747182896 p^{18} T^{25} + 512117152 p^{20} T^{26} - 56141240 p^{22} T^{27} - 7996188 p^{24} T^{28} - 27816 p^{26} T^{29} + 128 p^{28} T^{30} + 16 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
47 \( 1 - 20438982 T^{4} + 282981411633661 T^{8} - \)\(27\!\cdots\!94\)\( T^{12} + \)\(21\!\cdots\!22\)\( T^{16} - \)\(14\!\cdots\!82\)\( T^{20} + \)\(84\!\cdots\!63\)\( T^{24} - \)\(45\!\cdots\!18\)\( T^{28} + \)\(22\!\cdots\!58\)\( T^{32} - \)\(45\!\cdots\!18\)\( p^{8} T^{36} + \)\(84\!\cdots\!63\)\( p^{16} T^{40} - \)\(14\!\cdots\!82\)\( p^{24} T^{44} + \)\(21\!\cdots\!22\)\( p^{32} T^{48} - \)\(27\!\cdots\!94\)\( p^{40} T^{52} + 282981411633661 p^{48} T^{56} - 20438982 p^{56} T^{60} + p^{64} T^{64} \)
53 \( ( 1 - 68 T + 2312 T^{2} + 65572 T^{3} - 8064740 T^{4} + 35964812 T^{5} + 18349915256 T^{6} + 322510949524 T^{7} - 205995415393532 T^{8} + 9426150300049660 T^{9} + 5064897381427944 T^{10} - 24057793530563467868 T^{11} + \)\(60\!\cdots\!92\)\( T^{12} + \)\(51\!\cdots\!68\)\( T^{13} - \)\(22\!\cdots\!64\)\( T^{14} - \)\(14\!\cdots\!08\)\( T^{15} + \)\(20\!\cdots\!82\)\( T^{16} - \)\(14\!\cdots\!08\)\( p^{2} T^{17} - \)\(22\!\cdots\!64\)\( p^{4} T^{18} + \)\(51\!\cdots\!68\)\( p^{6} T^{19} + \)\(60\!\cdots\!92\)\( p^{8} T^{20} - 24057793530563467868 p^{10} T^{21} + 5064897381427944 p^{12} T^{22} + 9426150300049660 p^{14} T^{23} - 205995415393532 p^{16} T^{24} + 322510949524 p^{18} T^{25} + 18349915256 p^{20} T^{26} + 35964812 p^{22} T^{27} - 8064740 p^{24} T^{28} + 65572 p^{26} T^{29} + 2312 p^{28} T^{30} - 68 p^{30} T^{31} + p^{32} T^{32} )^{2} \)
59 \( ( 1 - 34236 T^{2} + 593396164 T^{4} - 6850249279428 T^{6} + 58712438660781428 T^{8} - \)\(39\!\cdots\!60\)\( T^{10} + \)\(21\!\cdots\!20\)\( T^{12} - \)\(98\!\cdots\!32\)\( T^{14} + \)\(37\!\cdots\!86\)\( T^{16} - \)\(98\!\cdots\!32\)\( p^{4} T^{18} + \)\(21\!\cdots\!20\)\( p^{8} T^{20} - \)\(39\!\cdots\!60\)\( p^{12} T^{22} + 58712438660781428 p^{16} T^{24} - 6850249279428 p^{20} T^{26} + 593396164 p^{24} T^{28} - 34236 p^{28} T^{30} + p^{32} T^{32} )^{2} \)
61 \( ( 1 - 33120 T^{2} + 570455096 T^{4} - 6668554929696 T^{6} + 58764621079937820 T^{8} - \)\(41\!\cdots\!64\)\( T^{10} + \)\(23\!\cdots\!48\)\( T^{12} - \)\(11\!\cdots\!36\)\( T^{14} + \)\(46\!\cdots\!22\)\( T^{16} - \)\(11\!\cdots\!36\)\( p^{4} T^{18} + \)\(23\!\cdots\!48\)\( p^{8} T^{20} - \)\(41\!\cdots\!64\)\( p^{12} T^{22} + 58764621079937820 p^{16} T^{24} - 6668554929696 p^{20} T^{26} + 570455096 p^{24} T^{28} - 33120 p^{28} T^{30} + p^{32} T^{32} )^{2} \)
67 \( 1 - 32T + 512T^{2} + 1.24e6T^{3} - 1.32e8T^{4} + 7.76e7T^{5} + 8.44e11T^{6} - 1.48e14T^{7} + 4.15e15T^{8} + 5.91e17T^{9} - 9.49e19T^{10} + 5.61e21T^{11} + 2.70e23T^{12} - 5.36e25T^{13} + 3.62e27T^{14} + 4.47e28T^{15} - 2.70e31T^{16} + 1.85e33T^{17} - 8.84e33T^{18} - 1.08e37T^{19} + 8.91e38T^{20} - 9.55e39T^{21} - 3.86e42T^{22} + 3.68e44T^{23} - 6.76e45T^{24} - 1.37e48T^{25} + 1.33e50T^{26} - 3.56e51T^{27} - 4.32e53T^{28} + 4.69e55T^{29} - 1.39e57T^{30} - 1.13e59T^{31}+O(T^{32}) \)
71 \( 1 - 1.64e4T^{2} + 2.25e8T^{4} - 1.82e12T^{6} + 1.58e16T^{8} - 1.07e20T^{10} + 8.89e23T^{12} - 5.49e27T^{14} + 3.89e31T^{16} - 2.11e35T^{18} + 1.44e39T^{20} - 7.68e42T^{22} + 4.93e46T^{24} - 2.36e50T^{26} + 1.44e54T^{28} - 6.77e57T^{30}+O(T^{32}) \)
73 \( 1 + 5.48e6T^{4} - 9.54e14T^{8} + 5.41e22T^{12} + 5.59e29T^{16} - 6.64e37T^{20} + 1.06e45T^{24} + 1.94e52T^{28}+O(T^{31}) \)
79 \( 1 - 1.06e5T^{2} + 5.63e9T^{4} - 1.95e14T^{6} + 5.05e18T^{8} - 1.04e23T^{10} + 1.80e27T^{12} - 2.68e31T^{14} + 3.50e35T^{16} - 4.09e39T^{18} + 4.31e43T^{20} - 4.15e47T^{22} + 3.66e51T^{24} - 2.98e55T^{26} + 2.24e59T^{28} - 1.56e63T^{30}+O(T^{31}) \)
83 \( 1 + 1.59e8T^{4} + 1.43e16T^{8} + 1.12e24T^{12} + 8.33e31T^{16} + 5.39e39T^{20} + 3.10e47T^{24} + 1.66e55T^{28}+O(T^{31}) \)
89 \( 1 - 1.90e5T^{2} + 1.78e10T^{4} - 1.08e15T^{6} + 4.86e19T^{8} - 1.70e24T^{10} + 4.86e28T^{12} - 1.16e33T^{14} + 2.40e37T^{16} - 4.28e41T^{18} + 6.74e45T^{20} - 9.41e49T^{22} + 1.17e54T^{24} - 1.32e58T^{26} + 1.33e62T^{28}+O(T^{30}) \)
97 \( 1 + 4.67e8T^{4} + 1.06e17T^{8} + 1.61e25T^{12} + 1.86e33T^{16} + 1.72e41T^{20} + 1.32e49T^{24} + 8.88e56T^{28}+O(T^{30}) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{64} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.12218359129496710877431917135, −2.05583610812269382976837444511, −2.05323603106399761578537988229, −2.05173488731542428434809544692, −2.01722566978591461249623104488, −1.92117655662520040609441087917, −1.91287279980020510431881795961, −1.84102340375738935512747390263, −1.80523221726141990836538443545, −1.80378762822059224639245823142, −1.60782938597115253951264239897, −1.50965143082078983766904927878, −1.49631843993225844593720479698, −1.14133172483907125353069157791, −1.08012958994623367827963258005, −1.02825871201042733956365361506, −1.00475337420811099762516807647, −0.897010229355952556639695721454, −0.895985910911315652361091962011, −0.862105086905449307468963163199, −0.65570629848420308370985754999, −0.61985528781100270133053909596, −0.27545523773686160388288353042, −0.13614703011569606050213431824, −0.00196158479429059965389806376, 0.00196158479429059965389806376, 0.13614703011569606050213431824, 0.27545523773686160388288353042, 0.61985528781100270133053909596, 0.65570629848420308370985754999, 0.862105086905449307468963163199, 0.895985910911315652361091962011, 0.897010229355952556639695721454, 1.00475337420811099762516807647, 1.02825871201042733956365361506, 1.08012958994623367827963258005, 1.14133172483907125353069157791, 1.49631843993225844593720479698, 1.50965143082078983766904927878, 1.60782938597115253951264239897, 1.80378762822059224639245823142, 1.80523221726141990836538443545, 1.84102340375738935512747390263, 1.91287279980020510431881795961, 1.92117655662520040609441087917, 2.01722566978591461249623104488, 2.05173488731542428434809544692, 2.05323603106399761578537988229, 2.05583610812269382976837444511, 2.12218359129496710877431917135

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

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