Properties

Degree $8$
Conductor $37822859361$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·5-s − 2·8-s + 8·10-s − 4·11-s − 16·13-s − 4·17-s − 12·20-s + 8·22-s − 4·23-s + 12·25-s + 32·26-s + 16·29-s − 8·31-s + 6·32-s + 8·34-s + 8·37-s + 8·40-s + 8·41-s − 12·44-s + 8·46-s − 24·50-s − 48·52-s − 4·53-s + 16·55-s − 32·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.78·5-s − 0.707·8-s + 2.52·10-s − 1.20·11-s − 4.43·13-s − 0.970·17-s − 2.68·20-s + 1.70·22-s − 0.834·23-s + 12/5·25-s + 6.27·26-s + 2.97·29-s − 1.43·31-s + 1.06·32-s + 1.37·34-s + 1.31·37-s + 1.26·40-s + 1.24·41-s − 1.80·44-s + 1.17·46-s − 3.39·50-s − 6.65·52-s − 0.549·53-s + 2.15·55-s − 4.20·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.195332\)
\(L(\frac12)\) \(\approx\) \(0.195332\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$D_4\times C_2$ \( 1 + p T + T^{2} - p T^{3} - 3 T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
5$D_4\times C_2$ \( 1 + 4 T + 4 T^{2} + 8 T^{3} + 39 T^{4} + 8 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_{4}$ \( ( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 56 p T^{5} - 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^3$ \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 4 T - 2 T^{2} - 112 T^{3} - 573 T^{4} - 112 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 8 T - 6 T^{2} + 64 T^{3} + 1955 T^{4} + 64 p T^{5} - 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 8 T - 62 T^{2} - 64 T^{3} + 8619 T^{4} - 64 p T^{5} - 62 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 16 T + 88 T^{2} - 736 T^{3} + 8887 T^{4} - 736 p T^{5} + 88 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^3$ \( 1 - 102 T^{2} + 5915 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 8 T + 656 T^{3} - 5905 T^{4} + 656 p T^{5} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 512 p T^{5} + 66 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 20 T + 140 T^{2} - 1640 T^{3} + 24079 T^{4} - 1640 p T^{5} + 140 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 8 T + 208 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−8.042084250611511671050344033018, −7.976962911792102249882217834644, −7.66915207771509112154498116576, −7.24270604105199894833649870860, −7.21090942184303427356446428880, −6.94689985994893705683983160196, −6.83791026211567828557377316255, −6.78546102473409594983307933964, −6.08411675096087786027221507372, −5.86041074892827605199304610430, −5.47734263462445795433573042490, −5.12590229259933459034359106841, −4.84796881351525067302331777052, −4.74158837045971618014781920286, −4.49463792035997144237331156952, −4.13591865704040257148115596797, −4.01591178117714525488973306410, −3.34012287377486296780811411006, −2.75319720552970535818630212536, −2.74363793057589466798992633355, −2.43344063322604878847884816356, −2.36156477598132384456261576151, −1.63312452912655866259965414681, −0.74835705711905262724000224294, −0.28796859418613877262425101429, 0.28796859418613877262425101429, 0.74835705711905262724000224294, 1.63312452912655866259965414681, 2.36156477598132384456261576151, 2.43344063322604878847884816356, 2.74363793057589466798992633355, 2.75319720552970535818630212536, 3.34012287377486296780811411006, 4.01591178117714525488973306410, 4.13591865704040257148115596797, 4.49463792035997144237331156952, 4.74158837045971618014781920286, 4.84796881351525067302331777052, 5.12590229259933459034359106841, 5.47734263462445795433573042490, 5.86041074892827605199304610430, 6.08411675096087786027221507372, 6.78546102473409594983307933964, 6.83791026211567828557377316255, 6.94689985994893705683983160196, 7.21090942184303427356446428880, 7.24270604105199894833649870860, 7.66915207771509112154498116576, 7.976962911792102249882217834644, 8.042084250611511671050344033018

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