Properties

Label 8-882e4-1.1-c1e4-0-12
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $2460.26$
Root an. cond. $2.65382$
Motivic weight $1$
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  + 4·2-s + 2·3-s + 10·4-s + 2·5-s + 8·6-s + 20·8-s + 3·9-s + 8·10-s − 4·11-s + 20·12-s + 4·15-s + 35·16-s + 4·17-s + 12·18-s + 10·19-s + 20·20-s − 16·22-s + 2·23-s + 40·24-s + 5·25-s + 10·27-s + 4·29-s + 16·30-s − 24·31-s + 56·32-s − 8·33-s + 16·34-s + ⋯
L(s)  = 1  + 2.82·2-s + 1.15·3-s + 5·4-s + 0.894·5-s + 3.26·6-s + 7.07·8-s + 9-s + 2.52·10-s − 1.20·11-s + 5.77·12-s + 1.03·15-s + 35/4·16-s + 0.970·17-s + 2.82·18-s + 2.29·19-s + 4.47·20-s − 3.41·22-s + 0.417·23-s + 8.16·24-s + 25-s + 1.92·27-s + 0.742·29-s + 2.92·30-s − 4.31·31-s + 9.89·32-s − 1.39·33-s + 2.74·34-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2460.26\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(51.07364612\)
\(L(\frac12)\) \(\approx\) \(51.07364612\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 - 2 T - T^{2} + 2 p T^{3} - 4 p T^{4} + 2 p^{2} T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^3$ \( 1 - 2 T^{2} - 165 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 10 T + 43 T^{2} - 10 p T^{3} + 52 p T^{4} - 10 p^{2} T^{5} + 43 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 4 T - 22 T^{2} + 80 T^{3} + 139 T^{4} + 80 p T^{5} - 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 + 4 T + 34 T^{2} - 368 T^{3} - 1637 T^{4} - 368 p T^{5} + 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^3$ \( 1 + 14 T^{2} - 1485 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 80 p T^{5} - 50 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 12 T + 26 T^{2} - 144 T^{3} + 3483 T^{4} - 144 p T^{5} + 26 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
61$D_{4}$ \( ( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 10 T + 143 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 4 T - 110 T^{2} - 80 T^{3} + 9379 T^{4} - 80 p T^{5} - 110 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 6 T + 143 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 24 T + 278 T^{2} + 2880 T^{3} + 29619 T^{4} + 2880 p T^{5} + 278 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 4 T - 158 T^{2} - 80 T^{3} + 19315 T^{4} - 80 p T^{5} - 158 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.31034161089436904541696375998, −6.83262111191519523933700334965, −6.81281827754723879696805369568, −6.77343874263190645450826834328, −5.99303144125706068855588775838, −5.94105885439756850034183886156, −5.88791524679931816619463832628, −5.51021321876497266921793463478, −5.44772261471286630358558427001, −5.13005932000762808024449690373, −4.92680244568265268925859438190, −4.68383518381932287360637471058, −4.57123502161400594395656962633, −4.22220973266364490290936856149, −3.65594481165534648506263846435, −3.63831447976259961942917756773, −3.30374751120496943846137109414, −3.09880997328276191471419829720, −2.94348281852844384207722159660, −2.86160507481221046413497990912, −2.21852798134589504987379615172, −2.01837700243424224814722347538, −1.69695934251557328543135873910, −1.43098205278338399079499359255, −0.864340788097194765073690537970, 0.864340788097194765073690537970, 1.43098205278338399079499359255, 1.69695934251557328543135873910, 2.01837700243424224814722347538, 2.21852798134589504987379615172, 2.86160507481221046413497990912, 2.94348281852844384207722159660, 3.09880997328276191471419829720, 3.30374751120496943846137109414, 3.63831447976259961942917756773, 3.65594481165534648506263846435, 4.22220973266364490290936856149, 4.57123502161400594395656962633, 4.68383518381932287360637471058, 4.92680244568265268925859438190, 5.13005932000762808024449690373, 5.44772261471286630358558427001, 5.51021321876497266921793463478, 5.88791524679931816619463832628, 5.94105885439756850034183886156, 5.99303144125706068855588775838, 6.77343874263190645450826834328, 6.81281827754723879696805369568, 6.83262111191519523933700334965, 7.31034161089436904541696375998

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