Properties

Label 8-210e4-1.1-c1e4-0-6
Degree $8$
Conductor $1944810000$
Sign $1$
Analytic cond. $7.90652$
Root an. cond. $1.29493$
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  − 2·2-s − 2·3-s + 4-s − 6·5-s + 4·6-s − 10·7-s + 2·8-s − 3·9-s + 12·10-s − 9·11-s − 2·12-s − 8·13-s + 20·14-s + 12·15-s − 4·16-s + 6·18-s − 12·19-s − 6·20-s + 20·21-s + 18·22-s − 3·23-s − 4·24-s + 17·25-s + 16·26-s + 14·27-s − 10·28-s − 24·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s − 2.68·5-s + 1.63·6-s − 3.77·7-s + 0.707·8-s − 9-s + 3.79·10-s − 2.71·11-s − 0.577·12-s − 2.21·13-s + 5.34·14-s + 3.09·15-s − 16-s + 1.41·18-s − 2.75·19-s − 1.34·20-s + 4.36·21-s + 3.83·22-s − 0.625·23-s − 0.816·24-s + 17/5·25-s + 3.13·26-s + 2.69·27-s − 1.88·28-s − 4.38·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(7.90652\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{210} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad2$C_2$ \( ( 1 + T + T^{2} )^{2} \)
3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
good11$D_4\times C_2$ \( 1 + 9 T + 53 T^{2} + 234 T^{3} + 852 T^{4} + 234 p T^{5} + 53 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
17$C_2^3$ \( 1 - 10 T^{2} - 189 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
23$D_4\times C_2$ \( 1 + 3 T - 31 T^{2} - 18 T^{3} + 864 T^{4} - 18 p T^{5} - 31 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 396 p T^{5} + 71 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 6 T - 10 T^{2} - 132 T^{3} - 441 T^{4} - 132 p T^{5} - 10 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 9 T + 94 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 49 T^{2} + 660 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 18 T + 218 T^{2} + 1980 T^{3} + 14967 T^{4} + 1980 p T^{5} + 218 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 3 T - 91 T^{2} + 18 T^{3} + 6714 T^{4} + 18 p T^{5} - 91 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 9 T + 17 T^{2} + 486 T^{3} - 4164 T^{4} + 486 p T^{5} + 17 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 4266 p T^{5} + 401 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 9 T + 143 T^{2} - 1044 T^{3} + 10776 T^{4} - 1044 p T^{5} + 143 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 208 T^{2} + 19710 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + T - 83 T^{2} - 74 T^{3} + 736 T^{4} - 74 p T^{5} - 83 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 190 T^{2} + 18051 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 3 T - 163 T^{2} - 18 T^{3} + 20862 T^{4} - 18 p T^{5} - 163 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 13 T + 162 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−9.839743826847223911230454261298, −9.269536211777992615698852127849, −9.112732912108445086653620975119, −8.591566285069870119259010426548, −8.482153561432475369085288043251, −8.298034353305992430478819453414, −8.016148392298045080854207309545, −7.81072831076801319433178636120, −7.68250450267913797898968174621, −7.07908580228187458124869971229, −6.70062517645664319229962296092, −6.70008915606101660396625301212, −6.69780056937169874837460099749, −6.26906157516024104207835137048, −5.59276707497835537283545876510, −5.52451896649878383805904862502, −5.15254909529325554717854524016, −4.76561723960312156697153618089, −4.42826215055449403060272632859, −4.16883934413190939542220972389, −3.51328939468482384673657287617, −3.38968513791770543190322526023, −2.89464076198083253935105012705, −2.59809620727508189996851675768, −2.44311815692053573669502230748, 0, 0, 0, 0, 2.44311815692053573669502230748, 2.59809620727508189996851675768, 2.89464076198083253935105012705, 3.38968513791770543190322526023, 3.51328939468482384673657287617, 4.16883934413190939542220972389, 4.42826215055449403060272632859, 4.76561723960312156697153618089, 5.15254909529325554717854524016, 5.52451896649878383805904862502, 5.59276707497835537283545876510, 6.26906157516024104207835137048, 6.69780056937169874837460099749, 6.70008915606101660396625301212, 6.70062517645664319229962296092, 7.07908580228187458124869971229, 7.68250450267913797898968174621, 7.81072831076801319433178636120, 8.016148392298045080854207309545, 8.298034353305992430478819453414, 8.482153561432475369085288043251, 8.591566285069870119259010426548, 9.112732912108445086653620975119, 9.269536211777992615698852127849, 9.839743826847223911230454261298

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