Properties

Label 8-637e4-1.1-c1e4-0-23
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 3·3-s + 17·4-s − 3·5-s + 18·6-s + 30·8-s + 7·9-s − 18·10-s + 3·11-s + 51·12-s − 10·13-s − 9·15-s + 40·16-s + 12·17-s + 42·18-s − 3·19-s − 51·20-s + 18·22-s + 90·24-s + 11·25-s − 60·26-s + 18·27-s − 3·29-s − 54·30-s − 4·31-s + 54·32-s + 9·33-s + ⋯
L(s)  = 1  + 4.24·2-s + 1.73·3-s + 17/2·4-s − 1.34·5-s + 7.34·6-s + 10.6·8-s + 7/3·9-s − 5.69·10-s + 0.904·11-s + 14.7·12-s − 2.77·13-s − 2.32·15-s + 10·16-s + 2.91·17-s + 9.89·18-s − 0.688·19-s − 11.4·20-s + 3.83·22-s + 18.3·24-s + 11/5·25-s − 11.7·26-s + 3.46·27-s − 0.557·29-s − 9.85·30-s − 0.718·31-s + 9.54·32-s + 1.56·33-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: \(0\)
Selberg data: \((8,\ 7^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(48.80060144\)
\(L(\frac12)\) \(\approx\) \(48.80060144\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
3$D_4\times C_2$ \( 1 - p T + 2 T^{2} - p T^{3} + 13 T^{4} - p^{2} T^{5} + 2 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
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$D_4\times C_2$ \( 1 - 3 T - 4 T^{2} + 27 T^{3} - 51 T^{4} + 27 p T^{5} - 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 + 3 T - 20 T^{2} - 27 T^{3} + 309 T^{4} - 27 p T^{5} - 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
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$C_2^2$ \( ( 1 + 113 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 12 T + 19 T^{2} + 108 T^{3} + 1488 T^{4} + 108 p T^{5} + 19 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
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$D_{4}$ \( ( 1 - 21 T + 257 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \)
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

−7.62349574484209797396147154989, −7.10702912309122237467915676185, −7.04822610718780718941328056415, −6.91256867215572089599375996086, −6.79252910823150672822878048793, −6.13898392007460558667063029527, −6.12940927769487844689010925985, −5.65800520588285851243097956677, −5.57029944182178105596647264099, −5.08102241373909164256810080829, −4.86545141379106381188143809306, −4.83890177211702428768172137540, −4.79403713733497976918580727950, −4.26394807553888105562467909145, −4.21395319475268831688290622089, −3.82910867323253268922916268207, −3.62992044980620075378954273268, −3.47515071161759657347342510798, −3.25070604956292780961921631338, −2.89759278150173823882256761019, −2.81402447053182029100347526799, −2.32351608880879227192380520532, −2.03914034028436336896142654439, −1.32201192923907521844642456413, −0.908292234904990279633198491028, 0.908292234904990279633198491028, 1.32201192923907521844642456413, 2.03914034028436336896142654439, 2.32351608880879227192380520532, 2.81402447053182029100347526799, 2.89759278150173823882256761019, 3.25070604956292780961921631338, 3.47515071161759657347342510798, 3.62992044980620075378954273268, 3.82910867323253268922916268207, 4.21395319475268831688290622089, 4.26394807553888105562467909145, 4.79403713733497976918580727950, 4.83890177211702428768172137540, 4.86545141379106381188143809306, 5.08102241373909164256810080829, 5.57029944182178105596647264099, 5.65800520588285851243097956677, 6.12940927769487844689010925985, 6.13898392007460558667063029527, 6.79252910823150672822878048793, 6.91256867215572089599375996086, 7.04822610718780718941328056415, 7.10702912309122237467915676185, 7.62349574484209797396147154989

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