Properties

Label 8-637e4-1.1-c1e4-0-26
Degree $8$
Conductor $164648481361$
Sign $1$
Analytic cond. $669.369$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 6·3-s + 5·4-s − 3·5-s + 18·6-s − 6·8-s + 13·9-s + 9·10-s − 6·11-s − 30·12-s − 10·13-s + 18·15-s + 4·16-s − 6·17-s − 39·18-s + 6·19-s − 15·20-s + 18·22-s + 36·24-s + 11·25-s + 30·26-s − 3·29-s − 54·30-s − 4·31-s + 36·33-s + 18·34-s + 65·36-s + ⋯
L(s)  = 1  − 2.12·2-s − 3.46·3-s + 5/2·4-s − 1.34·5-s + 7.34·6-s − 2.12·8-s + 13/3·9-s + 2.84·10-s − 1.80·11-s − 8.66·12-s − 2.77·13-s + 4.64·15-s + 16-s − 1.45·17-s − 9.19·18-s + 1.37·19-s − 3.35·20-s + 3.83·22-s + 7.34·24-s + 11/5·25-s + 5.88·26-s − 0.557·29-s − 9.85·30-s − 0.718·31-s + 6.26·33-s + 3.08·34-s + 65/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(7^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(669.369\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 7^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
good2$C_2^2$$\times$$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \)
3$D_{4}$ \( ( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \)
5$D_4\times C_2$ \( 1 + 3 T - 2 T^{2} + 3 T^{3} + 51 T^{4} + 3 p T^{5} - 2 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 6 T + 13 T^{2} - 66 T^{3} - 372 T^{4} - 66 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 26 T^{2} + 147 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 3 T - 20 T^{2} - 3 p T^{3} - 9 p T^{4} - 3 p^{2} T^{5} - 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 + 4 T - 5 T^{2} - 164 T^{3} - 1016 T^{4} - 164 p T^{5} - 5 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 6 T - 50 T^{2} + 24 T^{3} + 4239 T^{4} + 24 p T^{5} - 50 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 5 T + 34 T^{2} - 475 T^{3} - 2843 T^{4} - 475 p T^{5} + 34 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^3$ \( 1 - 89 T^{2} + 5712 T^{4} - 89 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 12 T + 7 T^{2} + 372 T^{3} + 8328 T^{4} + 372 p T^{5} + 7 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 113 T^{2} + 9288 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
67$D_{4}$ \( ( 1 + 12 T + 125 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_4\times C_2$ \( 1 - 6 T + 10 T^{2} + 696 T^{3} - 6921 T^{4} + 696 p T^{5} + 10 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 21 T + 184 T^{2} + 1659 T^{3} + 18879 T^{4} + 1659 p T^{5} + 184 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 31 T + 538 T^{2} + 7099 T^{3} + 76303 T^{4} + 7099 p T^{5} + 538 p^{2} T^{6} + 31 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.242991401730279972782828780390, −7.62520217637335834017986954306, −7.54322386340171729115201190047, −7.38729008597001431775717312410, −7.27421483754488196004475840244, −6.93190067328791730404683455113, −6.83450220255063357740806204950, −6.50999378888106660827924594467, −6.27242090166650313935153716975, −5.97710591948254184475290006370, −5.62005578126017163805885598093, −5.53756554414450238262609527407, −5.24514147000313573403109994682, −4.97042761414765212222222217190, −4.95170586053693634784177388736, −4.60954723869384635520296905504, −4.47283315213456736615072293224, −4.09147235014513230646258640690, −3.16748509140987734906855591612, −3.11221270724253022948343790276, −2.94912396629309653339228610913, −2.59550403565865417696718474522, −2.00527308999423146027154061936, −1.61714557940711332385785988550, −1.00929791706032927283141065071, 0, 0, 0, 0, 1.00929791706032927283141065071, 1.61714557940711332385785988550, 2.00527308999423146027154061936, 2.59550403565865417696718474522, 2.94912396629309653339228610913, 3.11221270724253022948343790276, 3.16748509140987734906855591612, 4.09147235014513230646258640690, 4.47283315213456736615072293224, 4.60954723869384635520296905504, 4.95170586053693634784177388736, 4.97042761414765212222222217190, 5.24514147000313573403109994682, 5.53756554414450238262609527407, 5.62005578126017163805885598093, 5.97710591948254184475290006370, 6.27242090166650313935153716975, 6.50999378888106660827924594467, 6.83450220255063357740806204950, 6.93190067328791730404683455113, 7.27421483754488196004475840244, 7.38729008597001431775717312410, 7.54322386340171729115201190047, 7.62520217637335834017986954306, 8.242991401730279972782828780390

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