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Properties

Label 8-162e4-1.1-c11e4-0-1
Degree $8$
Conductor $688747536$
Sign $1$
Analytic cond. $2.40038\times 10^{8}$
Root an. cond. $11.1566$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 64·2-s + 1.02e3·4-s + 4.90e3·5-s − 1.12e4·7-s + 6.55e4·8-s − 3.14e5·10-s − 4.51e5·11-s + 9.04e5·13-s + 7.19e5·14-s − 4.19e6·16-s + 1.29e7·17-s + 2.40e7·19-s + 5.02e6·20-s + 2.89e7·22-s − 1.58e7·23-s + 8.49e6·25-s − 5.78e7·26-s − 1.15e7·28-s + 3.15e7·29-s − 6.32e7·31-s + 6.71e7·32-s − 8.30e8·34-s − 5.51e7·35-s + 1.24e9·37-s − 1.53e9·38-s + 3.21e8·40-s − 2.06e9·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 0.702·5-s − 0.252·7-s + 0.707·8-s − 0.993·10-s − 0.845·11-s + 0.675·13-s + 0.357·14-s − 16-s + 2.21·17-s + 2.22·19-s + 0.351·20-s + 1.19·22-s − 0.513·23-s + 0.174·25-s − 0.955·26-s − 0.126·28-s + 0.285·29-s − 0.396·31-s + 0.353·32-s − 3.13·34-s − 0.177·35-s + 2.94·37-s − 3.14·38-s + 0.496·40-s − 2.77·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(2.40038\times 10^{8}\)
Root analytic conductor: \(11.1566\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{16} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\)

Particular Values

\(L(6)\) \(\approx\) \(0.3366574356\)
\(L(\frac12)\) \(\approx\) \(0.3366574356\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( ( 1 + p^{5} T + p^{10} T^{2} )^{2} \)
3 \( 1 \)
good5$D_4\times C_2$ \( 1 - 4908 T + 15591254 T^{2} + 87518513664 p T^{3} - 139527109779261 p^{2} T^{4} + 87518513664 p^{12} T^{5} + 15591254 p^{22} T^{6} - 4908 p^{33} T^{7} + p^{44} T^{8} \)
7$D_4\times C_2$ \( 1 + 1606 p T - 497020277 p T^{2} - 80100163814 p^{2} T^{3} + 180409923867434692 p^{2} T^{4} - 80100163814 p^{13} T^{5} - 497020277 p^{23} T^{6} + 1606 p^{34} T^{7} + p^{44} T^{8} \)
11$D_4\times C_2$ \( 1 + 451812 T - 406005858838 T^{2} + 17854074512939520 T^{3} + \)\(23\!\cdots\!19\)\( T^{4} + 17854074512939520 p^{11} T^{5} - 406005858838 p^{22} T^{6} + 451812 p^{33} T^{7} + p^{44} T^{8} \)
13$D_4\times C_2$ \( 1 - 904502 T - 1051495057487 T^{2} + 1550951264110048666 T^{3} - \)\(11\!\cdots\!36\)\( T^{4} + 1550951264110048666 p^{11} T^{5} - 1051495057487 p^{22} T^{6} - 904502 p^{33} T^{7} + p^{44} T^{8} \)
17$D_{4}$ \( ( 1 - 6491916 T + 3922113677762 p T^{2} - 6491916 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 - 12001558 T + 218545376621775 T^{2} - 12001558 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 15847332 T - 1255332024259582 T^{2} - \)\(63\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!59\)\( T^{4} - \)\(63\!\cdots\!36\)\( p^{11} T^{5} - 1255332024259582 p^{22} T^{6} + 15847332 p^{33} T^{7} + p^{44} T^{8} \)
29$D_4\times C_2$ \( 1 - 1089120 p T - 21986607387909322 T^{2} + \)\(15\!\cdots\!20\)\( p T^{3} + \)\(36\!\cdots\!43\)\( T^{4} + \)\(15\!\cdots\!20\)\( p^{12} T^{5} - 21986607387909322 p^{22} T^{6} - 1089120 p^{34} T^{7} + p^{44} T^{8} \)
31$D_4\times C_2$ \( 1 + 63255688 T + 29690336803345522 T^{2} - \)\(48\!\cdots\!20\)\( T^{3} - \)\(87\!\cdots\!41\)\( T^{4} - \)\(48\!\cdots\!20\)\( p^{11} T^{5} + 29690336803345522 p^{22} T^{6} + 63255688 p^{33} T^{7} + p^{44} T^{8} \)
37$D_{4}$ \( ( 1 - 622016230 T + 442475283279094035 T^{2} - 622016230 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 2061695040 T + 2277905016557852942 T^{2} + \)\(43\!\cdots\!40\)\( p T^{3} + \)\(74\!\cdots\!43\)\( p^{2} T^{4} + \)\(43\!\cdots\!40\)\( p^{12} T^{5} + 2277905016557852942 p^{22} T^{6} + 2061695040 p^{33} T^{7} + p^{44} T^{8} \)
43$D_4\times C_2$ \( 1 - 285672608 T - 596852661277369382 T^{2} + \)\(33\!\cdots\!44\)\( T^{3} - \)\(47\!\cdots\!81\)\( T^{4} + \)\(33\!\cdots\!44\)\( p^{11} T^{5} - 596852661277369382 p^{22} T^{6} - 285672608 p^{33} T^{7} + p^{44} T^{8} \)
47$D_4\times C_2$ \( 1 + 1144669908 T - 2574855926048217358 T^{2} - \)\(12\!\cdots\!72\)\( T^{3} + \)\(57\!\cdots\!83\)\( T^{4} - \)\(12\!\cdots\!72\)\( p^{11} T^{5} - 2574855926048217358 p^{22} T^{6} + 1144669908 p^{33} T^{7} + p^{44} T^{8} \)
53$D_{4}$ \( ( 1 - 6003302616 T + 19960640735811915994 T^{2} - 6003302616 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 10466877444 T + 32358061674574744010 T^{2} + \)\(17\!\cdots\!52\)\( T^{3} + \)\(17\!\cdots\!71\)\( T^{4} + \)\(17\!\cdots\!52\)\( p^{11} T^{5} + 32358061674574744010 p^{22} T^{6} + 10466877444 p^{33} T^{7} + p^{44} T^{8} \)
61$D_4\times C_2$ \( 1 - 9401475122 T + 8384327701637669977 T^{2} + \)\(66\!\cdots\!30\)\( T^{3} + \)\(66\!\cdots\!04\)\( T^{4} + \)\(66\!\cdots\!30\)\( p^{11} T^{5} + 8384327701637669977 p^{22} T^{6} - 9401475122 p^{33} T^{7} + p^{44} T^{8} \)
67$D_4\times C_2$ \( 1 - 8629979906 T - 4926461536991847539 T^{2} + \)\(14\!\cdots\!46\)\( T^{3} - \)\(17\!\cdots\!52\)\( T^{4} + \)\(14\!\cdots\!46\)\( p^{11} T^{5} - 4926461536991847539 p^{22} T^{6} - 8629979906 p^{33} T^{7} + p^{44} T^{8} \)
71$D_{4}$ \( ( 1 - 30866231016 T + \)\(65\!\cdots\!02\)\( T^{2} - 30866231016 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 38999769970 T + \)\(96\!\cdots\!63\)\( T^{2} - 38999769970 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 8802915578 T - \)\(14\!\cdots\!95\)\( T^{2} + \)\(10\!\cdots\!62\)\( T^{3} + \)\(15\!\cdots\!64\)\( T^{4} + \)\(10\!\cdots\!62\)\( p^{11} T^{5} - \)\(14\!\cdots\!95\)\( p^{22} T^{6} - 8802915578 p^{33} T^{7} + p^{44} T^{8} \)
83$D_4\times C_2$ \( 1 + 2103831336 T - \)\(57\!\cdots\!46\)\( T^{2} - \)\(42\!\cdots\!12\)\( T^{3} - \)\(13\!\cdots\!57\)\( T^{4} - \)\(42\!\cdots\!12\)\( p^{11} T^{5} - \)\(57\!\cdots\!46\)\( p^{22} T^{6} + 2103831336 p^{33} T^{7} + p^{44} T^{8} \)
89$D_{4}$ \( ( 1 + 59947387524 T + \)\(48\!\cdots\!18\)\( T^{2} + 59947387524 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 59716829222 T - \)\(56\!\cdots\!87\)\( T^{2} + \)\(30\!\cdots\!70\)\( T^{3} + \)\(88\!\cdots\!80\)\( T^{4} + \)\(30\!\cdots\!70\)\( p^{11} T^{5} - \)\(56\!\cdots\!87\)\( p^{22} T^{6} - 59716829222 p^{33} T^{7} + p^{44} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.31335948761095491666902217111, −7.28841061473511805042099726561, −6.94770023166833007011577128448, −6.58142807560751347038279352821, −6.30958173655257373097495503382, −5.92455800725991737499160725844, −5.70077867745050757677550342284, −5.35709479777994763154902387954, −5.15301366795962659828814139056, −5.04914978725081969122738384251, −4.82285778870164281348385592089, −3.89895537514619404969605642946, −3.79888689562668241802610035601, −3.60803608793739309174473214020, −3.59051185064717311499716888239, −2.82016305886151437433114306634, −2.50862426990483494070138726438, −2.32563026421510874534017372218, −2.28160375049001671333860198903, −1.43817023149310180850101052305, −1.17801894876843414281125874864, −1.15586812387611084936561482600, −0.78386009641322578619401438729, −0.73152075774113750338068220137, −0.07159493609718697416315868881, 0.07159493609718697416315868881, 0.73152075774113750338068220137, 0.78386009641322578619401438729, 1.15586812387611084936561482600, 1.17801894876843414281125874864, 1.43817023149310180850101052305, 2.28160375049001671333860198903, 2.32563026421510874534017372218, 2.50862426990483494070138726438, 2.82016305886151437433114306634, 3.59051185064717311499716888239, 3.60803608793739309174473214020, 3.79888689562668241802610035601, 3.89895537514619404969605642946, 4.82285778870164281348385592089, 5.04914978725081969122738384251, 5.15301366795962659828814139056, 5.35709479777994763154902387954, 5.70077867745050757677550342284, 5.92455800725991737499160725844, 6.30958173655257373097495503382, 6.58142807560751347038279352821, 6.94770023166833007011577128448, 7.28841061473511805042099726561, 7.31335948761095491666902217111

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