Properties

Label 8-75e4-1.1-c21e4-0-0
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $1.93032\times 10^{9}$
Root an. cond. $14.4778$
Motivic weight $21$
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  + 897·2-s + 2.36e5·3-s − 4.37e6·4-s + 2.11e8·6-s + 2.34e8·7-s − 4.81e9·8-s + 3.48e10·9-s + 3.14e10·11-s − 1.03e12·12-s + 2.70e10·13-s + 2.10e11·14-s + 6.70e12·16-s + 2.94e12·17-s + 3.12e13·18-s − 2.42e13·19-s + 5.54e13·21-s + 2.82e13·22-s − 1.03e13·23-s − 1.13e15·24-s + 2.42e13·26-s + 4.11e15·27-s − 1.02e15·28-s + 4.72e15·29-s − 1.09e15·31-s + 8.96e15·32-s + 7.43e15·33-s + 2.64e15·34-s + ⋯
L(s)  = 1  + 0.619·2-s + 2.30·3-s − 2.08·4-s + 1.43·6-s + 0.313·7-s − 1.58·8-s + 10/3·9-s + 0.366·11-s − 4.81·12-s + 0.0543·13-s + 0.194·14-s + 1.52·16-s + 0.354·17-s + 2.06·18-s − 0.908·19-s + 0.724·21-s + 0.226·22-s − 0.0520·23-s − 3.66·24-s + 0.0336·26-s + 3.84·27-s − 0.654·28-s + 2.08·29-s − 0.239·31-s + 1.40·32-s + 0.845·33-s + 0.219·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+21/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.93032\times 10^{9}\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :21/2, 21/2, 21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(4.546063225\)
\(L(\frac12)\) \(\approx\) \(4.546063225\)
\(L(\frac{23}{2})\) 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)$
bad3$C_1$ \( ( 1 - p^{10} T )^{4} \)
5 \( 1 \)
good2$C_2 \wr S_4$ \( 1 - 897 T + 2589085 p T^{2} - 117236669 p^{5} T^{3} + 14637391839 p^{10} T^{4} - 117236669 p^{26} T^{5} + 2589085 p^{43} T^{6} - 897 p^{63} T^{7} + p^{84} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 4787296 p^{2} T + 7592135803176028 p^{2} T^{2} + \)\(13\!\cdots\!36\)\( p^{4} T^{3} + \)\(73\!\cdots\!10\)\( p^{4} T^{4} + \)\(13\!\cdots\!36\)\( p^{25} T^{5} + 7592135803176028 p^{44} T^{6} - 4787296 p^{65} T^{7} + p^{84} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 31491830256 T + \)\(93\!\cdots\!92\)\( p T^{2} - \)\(46\!\cdots\!00\)\( p^{2} T^{3} + \)\(53\!\cdots\!06\)\( p^{3} T^{4} - \)\(46\!\cdots\!00\)\( p^{23} T^{5} + \)\(93\!\cdots\!92\)\( p^{43} T^{6} - 31491830256 p^{63} T^{7} + p^{84} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 27017977768 T - \)\(92\!\cdots\!72\)\( p T^{2} - \)\(33\!\cdots\!24\)\( p^{2} T^{3} + \)\(39\!\cdots\!30\)\( p^{3} T^{4} - \)\(33\!\cdots\!24\)\( p^{23} T^{5} - \)\(92\!\cdots\!72\)\( p^{43} T^{6} - 27017977768 p^{63} T^{7} + p^{84} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 2946095028888 T + \)\(67\!\cdots\!16\)\( p T^{2} - \)\(25\!\cdots\!20\)\( p^{2} T^{3} + \)\(19\!\cdots\!02\)\( p^{3} T^{4} - \)\(25\!\cdots\!20\)\( p^{23} T^{5} + \)\(67\!\cdots\!16\)\( p^{43} T^{6} - 2946095028888 p^{63} T^{7} + p^{84} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 24270353300752 T + \)\(48\!\cdots\!92\)\( p^{2} T^{2} + \)\(36\!\cdots\!36\)\( p^{3} T^{3} + \)\(21\!\cdots\!26\)\( p^{3} T^{4} + \)\(36\!\cdots\!36\)\( p^{24} T^{5} + \)\(48\!\cdots\!92\)\( p^{44} T^{6} + 24270353300752 p^{63} T^{7} + p^{84} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 10350924920928 T + \)\(70\!\cdots\!08\)\( T^{2} + \)\(70\!\cdots\!12\)\( T^{3} + \)\(23\!\cdots\!10\)\( T^{4} + \)\(70\!\cdots\!12\)\( p^{21} T^{5} + \)\(70\!\cdots\!08\)\( p^{42} T^{6} + 10350924920928 p^{63} T^{7} + p^{84} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 4728924677079096 T + \)\(11\!\cdots\!00\)\( T^{2} - \)\(17\!\cdots\!92\)\( T^{3} + \)\(17\!\cdots\!18\)\( T^{4} - \)\(17\!\cdots\!92\)\( p^{21} T^{5} + \)\(11\!\cdots\!00\)\( p^{42} T^{6} - 4728924677079096 p^{63} T^{7} + p^{84} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 1094923910405536 T + \)\(68\!\cdots\!48\)\( T^{2} + \)\(44\!\cdots\!48\)\( T^{3} + \)\(19\!\cdots\!54\)\( T^{4} + \)\(44\!\cdots\!48\)\( p^{21} T^{5} + \)\(68\!\cdots\!48\)\( p^{42} T^{6} + 1094923910405536 p^{63} T^{7} + p^{84} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 4813435696247096 T + \)\(17\!\cdots\!12\)\( T^{2} + \)\(33\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!50\)\( T^{4} + \)\(33\!\cdots\!36\)\( p^{21} T^{5} + \)\(17\!\cdots\!12\)\( p^{42} T^{6} + 4813435696247096 p^{63} T^{7} + p^{84} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 289731591445930344 T + \)\(49\!\cdots\!68\)\( T^{2} - \)\(57\!\cdots\!52\)\( T^{3} + \)\(55\!\cdots\!14\)\( T^{4} - \)\(57\!\cdots\!52\)\( p^{21} T^{5} + \)\(49\!\cdots\!68\)\( p^{42} T^{6} - 289731591445930344 p^{63} T^{7} + p^{84} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 451091912658458000 T + \)\(13\!\cdots\!00\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(44\!\cdots\!98\)\( T^{4} + \)\(27\!\cdots\!00\)\( p^{21} T^{5} + \)\(13\!\cdots\!00\)\( p^{42} T^{6} + 451091912658458000 p^{63} T^{7} + p^{84} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 813883435638492480 T + \)\(59\!\cdots\!20\)\( T^{2} + \)\(27\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!18\)\( T^{4} + \)\(27\!\cdots\!60\)\( p^{21} T^{5} + \)\(59\!\cdots\!20\)\( p^{42} T^{6} + 813883435638492480 p^{63} T^{7} + p^{84} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 697278335404085208 T + \)\(47\!\cdots\!28\)\( T^{2} + \)\(22\!\cdots\!72\)\( T^{3} + \)\(10\!\cdots\!50\)\( T^{4} + \)\(22\!\cdots\!72\)\( p^{21} T^{5} + \)\(47\!\cdots\!28\)\( p^{42} T^{6} + 697278335404085208 p^{63} T^{7} + p^{84} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 6622888614569598192 T + \)\(40\!\cdots\!72\)\( T^{2} - \)\(20\!\cdots\!04\)\( T^{3} + \)\(88\!\cdots\!34\)\( T^{4} - \)\(20\!\cdots\!04\)\( p^{21} T^{5} + \)\(40\!\cdots\!72\)\( p^{42} T^{6} - 6622888614569598192 p^{63} T^{7} + p^{84} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 7390887218011683320 T + \)\(30\!\cdots\!96\)\( T^{2} - \)\(27\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!46\)\( T^{4} - \)\(27\!\cdots\!80\)\( p^{21} T^{5} + \)\(30\!\cdots\!96\)\( p^{42} T^{6} - 7390887218011683320 p^{63} T^{7} + p^{84} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 24188188449376788688 T + \)\(66\!\cdots\!72\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!26\)\( T^{4} - \)\(12\!\cdots\!80\)\( p^{21} T^{5} + \)\(66\!\cdots\!72\)\( p^{42} T^{6} - 24188188449376788688 p^{63} T^{7} + p^{84} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 37390337803999713312 T + \)\(18\!\cdots\!88\)\( T^{2} + \)\(49\!\cdots\!64\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} + \)\(49\!\cdots\!64\)\( p^{21} T^{5} + \)\(18\!\cdots\!88\)\( p^{42} T^{6} + 37390337803999713312 p^{63} T^{7} + p^{84} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 37253672904265201432 T + \)\(53\!\cdots\!28\)\( T^{2} - \)\(13\!\cdots\!88\)\( T^{3} + \)\(10\!\cdots\!30\)\( T^{4} - \)\(13\!\cdots\!88\)\( p^{21} T^{5} + \)\(53\!\cdots\!28\)\( p^{42} T^{6} - 37253672904265201432 p^{63} T^{7} + p^{84} T^{8} \)
79$C_2 \wr S_4$ \( 1 + \)\(10\!\cdots\!00\)\( T + \)\(20\!\cdots\!16\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!46\)\( T^{4} + \)\(13\!\cdots\!00\)\( p^{21} T^{5} + \)\(20\!\cdots\!16\)\( p^{42} T^{6} + \)\(10\!\cdots\!00\)\( p^{63} T^{7} + p^{84} T^{8} \)
83$C_2 \wr S_4$ \( 1 - \)\(14\!\cdots\!96\)\( T + \)\(49\!\cdots\!60\)\( T^{2} - \)\(54\!\cdots\!56\)\( T^{3} + \)\(11\!\cdots\!26\)\( T^{4} - \)\(54\!\cdots\!56\)\( p^{21} T^{5} + \)\(49\!\cdots\!60\)\( p^{42} T^{6} - \)\(14\!\cdots\!96\)\( p^{63} T^{7} + p^{84} T^{8} \)
89$C_2 \wr S_4$ \( 1 - \)\(55\!\cdots\!48\)\( T + \)\(43\!\cdots\!52\)\( T^{2} - \)\(14\!\cdots\!96\)\( T^{3} + \)\(60\!\cdots\!14\)\( T^{4} - \)\(14\!\cdots\!96\)\( p^{21} T^{5} + \)\(43\!\cdots\!52\)\( p^{42} T^{6} - \)\(55\!\cdots\!48\)\( p^{63} T^{7} + p^{84} T^{8} \)
97$C_2 \wr S_4$ \( 1 - \)\(23\!\cdots\!28\)\( T + \)\(13\!\cdots\!32\)\( T^{2} - \)\(61\!\cdots\!20\)\( T^{3} + \)\(80\!\cdots\!66\)\( T^{4} - \)\(61\!\cdots\!20\)\( p^{21} T^{5} + \)\(13\!\cdots\!32\)\( p^{42} T^{6} - \)\(23\!\cdots\!28\)\( p^{63} T^{7} + p^{84} T^{8} \)
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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

−7.22639530576322043111833302287, −6.80735713819329200509574343002, −6.35386830467607815544680369770, −6.20575981451759787092153793599, −6.10036355139680912497850044115, −5.19237455553997743869634004930, −5.03779278453668928634461865772, −5.03634442319298137205168198625, −4.79878787603475283636371681302, −4.39238177739969696388592933140, −4.06310420032187211214261249356, −3.90478760883498069022513968770, −3.81276074114955752671402312844, −3.43366841103990700938123090627, −3.15877610266214529398002195473, −2.89478536220428508178481633179, −2.46829729830096611179535503780, −2.42694790482257175527798081887, −1.92671607588297497464029408750, −1.69130747885270809116816533325, −1.46592929815049956714237597212, −0.982133164506860285995281688304, −0.70407349876262103304756547841, −0.63422220520557733292249204716, −0.13215812560812576347308888797, 0.13215812560812576347308888797, 0.63422220520557733292249204716, 0.70407349876262103304756547841, 0.982133164506860285995281688304, 1.46592929815049956714237597212, 1.69130747885270809116816533325, 1.92671607588297497464029408750, 2.42694790482257175527798081887, 2.46829729830096611179535503780, 2.89478536220428508178481633179, 3.15877610266214529398002195473, 3.43366841103990700938123090627, 3.81276074114955752671402312844, 3.90478760883498069022513968770, 4.06310420032187211214261249356, 4.39238177739969696388592933140, 4.79878787603475283636371681302, 5.03634442319298137205168198625, 5.03779278453668928634461865772, 5.19237455553997743869634004930, 6.10036355139680912497850044115, 6.20575981451759787092153793599, 6.35386830467607815544680369770, 6.80735713819329200509574343002, 7.22639530576322043111833302287

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