# 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 of factors

## 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 + 64/7·14-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 = $$(64,\ 2^{32} \cdot 3^{32} \cdot 5^{32} \cdot 7^{32} ,\ ( \ : [1]^{32} ),\ 1 )$$ $$L(\frac{3}{2})$$ $$\approx$$ $$8.74116\times10^{-5}$$ $$L(\frac12)$$ $$\approx$$ $$8.74116\times10^{-5}$$ $$L(2)$$ 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 $$1 + 4.67e8T^{4} + 1.06e17T^{8} + 1.61e25T^{12} + 1.86e33T^{16} + 1.72e41T^{20} + 1.32e49T^{24} + 8.88e56T^{28}+O(T^{30})$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{64} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}