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Properties

Label 8-1520e4-1.1-c1e4-0-3
Degree $8$
Conductor $5.338\times 10^{12}$
Sign $1$
Analytic cond. $21701.1$
Root an. cond. $3.48385$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·5-s − 4·7-s − 4·11-s + 2·13-s + 8·15-s + 4·17-s − 4·19-s + 8·21-s + 8·23-s + 10·25-s + 6·27-s + 4·29-s − 4·31-s + 8·33-s + 16·35-s − 6·37-s − 4·39-s + 16·41-s − 4·43-s + 12·47-s + 4·49-s − 8·51-s − 10·53-s + 16·55-s + 8·57-s + 20·61-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s − 1.51·7-s − 1.20·11-s + 0.554·13-s + 2.06·15-s + 0.970·17-s − 0.917·19-s + 1.74·21-s + 1.66·23-s + 2·25-s + 1.15·27-s + 0.742·29-s − 0.718·31-s + 1.39·33-s + 2.70·35-s − 0.986·37-s − 0.640·39-s + 2.49·41-s − 0.609·43-s + 1.75·47-s + 4/7·49-s − 1.12·51-s − 1.37·53-s + 2.15·55-s + 1.05·57-s + 2.56·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(21701.1\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9975018949\)
\(L(\frac12)\) \(\approx\) \(0.9975018949\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{4} \)
19$C_1$ \( ( 1 + T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + 2 T + 4 T^{2} + 2 T^{3} + 2 T^{4} + 2 p T^{5} + 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
7$S_4\times C_2$ \( 1 + 4 T + 12 T^{2} + 36 T^{3} + 102 T^{4} + 36 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$\textrm{GL(2,3)}$ \( 1 + 4 T + 28 T^{2} + 100 T^{3} + 422 T^{4} + 100 p T^{5} + 28 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 2 T + 28 T^{2} - 46 T^{3} + 410 T^{4} - 46 p T^{5} + 28 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 4 T + 36 T^{2} - 188 T^{3} + 694 T^{4} - 188 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 8 T + 68 T^{2} - 376 T^{3} + 2358 T^{4} - 376 p T^{5} + 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$\textrm{GL(2,3)}$ \( 1 - 4 T + 84 T^{2} - 332 T^{3} + 3238 T^{4} - 332 p T^{5} + 84 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 4 T + 44 T^{2} - 140 T^{3} + 166 T^{4} - 140 p T^{5} + 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 6 T + 124 T^{2} + 626 T^{3} + 6442 T^{4} + 626 p T^{5} + 124 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 16 T + 220 T^{2} - 1936 T^{3} + 14438 T^{4} - 1936 p T^{5} + 220 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 4 T + 156 T^{2} + 468 T^{3} + 9750 T^{4} + 468 p T^{5} + 156 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 12 T + 124 T^{2} - 1036 T^{3} + 8294 T^{4} - 1036 p T^{5} + 124 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 10 T + 4 p T^{2} + 1406 T^{3} + 16506 T^{4} + 1406 p T^{5} + 4 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 172 T^{2} + 224 T^{3} + 13142 T^{4} + 224 p T^{5} + 172 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 20 T + 300 T^{2} - 2972 T^{3} + 26502 T^{4} - 2972 p T^{5} + 300 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 18 T + 276 T^{2} - 3130 T^{3} + 26930 T^{4} - 3130 p T^{5} + 276 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 20 T + 316 T^{2} - 3236 T^{3} + 30566 T^{4} - 3236 p T^{5} + 316 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 28 T + 548 T^{2} - 6916 T^{3} + 69526 T^{4} - 6916 p T^{5} + 548 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 16 T + 348 T^{2} - 3312 T^{3} + 40646 T^{4} - 3312 p T^{5} + 348 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 260 T^{2} + 112 T^{3} + 29862 T^{4} + 112 p T^{5} + 260 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 4 T + 212 T^{2} - 1244 T^{3} + 22134 T^{4} - 1244 p T^{5} + 212 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 30 T + 612 T^{2} - 8738 T^{3} + 98522 T^{4} - 8738 p T^{5} + 612 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.78640637620653242973433832133, −6.50879879594255589014961887892, −6.28135741490770889778501368230, −6.24411125598130606474362460943, −5.99499942851479253306723669856, −5.55465297024890703497978560209, −5.31219731892161718757037795716, −5.25075764706544746706626150008, −5.15871255454274716652356007146, −4.81012895611876814905395562315, −4.60266806887391510663125215378, −4.36632568048107215959714918484, −3.89867620458813527639279247162, −3.72529974220118549667073247480, −3.52045130597605910564229139364, −3.37776397168955146406808504636, −3.37589461550456262780437641586, −2.71631329295999927797377785765, −2.46917309036851989321327263069, −2.39338449353553854301783853164, −2.09957959018038073729488265120, −1.19671858130083446499615453632, −0.904002215318984343370097964583, −0.55766822594130104201468725597, −0.45241347681416962754148785768, 0.45241347681416962754148785768, 0.55766822594130104201468725597, 0.904002215318984343370097964583, 1.19671858130083446499615453632, 2.09957959018038073729488265120, 2.39338449353553854301783853164, 2.46917309036851989321327263069, 2.71631329295999927797377785765, 3.37589461550456262780437641586, 3.37776397168955146406808504636, 3.52045130597605910564229139364, 3.72529974220118549667073247480, 3.89867620458813527639279247162, 4.36632568048107215959714918484, 4.60266806887391510663125215378, 4.81012895611876814905395562315, 5.15871255454274716652356007146, 5.25075764706544746706626150008, 5.31219731892161718757037795716, 5.55465297024890703497978560209, 5.99499942851479253306723669856, 6.24411125598130606474362460943, 6.28135741490770889778501368230, 6.50879879594255589014961887892, 6.78640637620653242973433832133

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