Properties

Label 8-435e4-1.1-c1e4-0-7
Degree $8$
Conductor $35806100625$
Sign $1$
Analytic cond. $145.567$
Root an. cond. $1.86373$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 4·3-s + 3·4-s − 4·5-s + 12·6-s + 2·7-s − 2·8-s + 10·9-s + 12·10-s − 2·11-s − 12·12-s − 8·13-s − 6·14-s + 16·15-s + 4·16-s − 10·17-s − 30·18-s − 2·19-s − 12·20-s − 8·21-s + 6·22-s − 12·23-s + 8·24-s + 10·25-s + 24·26-s − 20·27-s + 6·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.30·3-s + 3/2·4-s − 1.78·5-s + 4.89·6-s + 0.755·7-s − 0.707·8-s + 10/3·9-s + 3.79·10-s − 0.603·11-s − 3.46·12-s − 2.21·13-s − 1.60·14-s + 4.13·15-s + 16-s − 2.42·17-s − 7.07·18-s − 0.458·19-s − 2.68·20-s − 1.74·21-s + 1.27·22-s − 2.50·23-s + 1.63·24-s + 2·25-s + 4.70·26-s − 3.84·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{4} \cdot 29^{4}\)
Sign: $1$
Analytic conductor: \(145.567\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 5^{4} \cdot 29^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad3$C_1$ \( ( 1 + T )^{4} \)
5$C_1$ \( ( 1 + T )^{4} \)
29$C_1$ \( ( 1 + T )^{4} \)
good2$C_4\wr C_2$ \( 1 + 3 T + 3 p T^{2} + 11 T^{3} + 17 T^{4} + 11 p T^{5} + 3 p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 13 T^{2} - 46 T^{3} + 88 T^{4} - 46 p T^{5} + 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 29 T^{2} + 70 T^{3} + 400 T^{4} + 70 p T^{5} + 29 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 55 T^{2} + 220 T^{3} + 928 T^{4} + 220 p T^{5} + 55 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 89 T^{2} + 470 T^{3} + 2332 T^{4} + 470 p T^{5} + 89 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 36 T^{2} + 58 T^{3} + 630 T^{4} + 58 p T^{5} + 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 80 T^{2} + 380 T^{3} + 1598 T^{4} + 380 p T^{5} + 80 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 64 T^{2} + 404 T^{3} + 2110 T^{4} + 404 p T^{5} + 64 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 160 T^{2} + 1040 T^{3} + 6478 T^{4} + 1040 p T^{5} + 160 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 84 T^{2} + 340 T^{3} + 1910 T^{4} + 340 p T^{5} + 84 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 56 T^{2} - 66 T^{3} + 2334 T^{4} - 66 p T^{5} + 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 187 T^{2} + 1620 T^{3} + 13096 T^{4} + 1620 p T^{5} + 187 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 132 T^{2} + 990 T^{3} + 8774 T^{4} + 990 p T^{5} + 132 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 24 T^{2} - 226 T^{3} + 6366 T^{4} - 226 p T^{5} + 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 26 T + 316 T^{2} + 2622 T^{3} + 19766 T^{4} + 2622 p T^{5} + 316 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 161 T^{2} - 694 T^{3} + 12472 T^{4} - 694 p T^{5} + 161 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 240 T^{2} + 1810 T^{3} + 24062 T^{4} + 1810 p T^{5} + 240 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 208 T^{2} + 320 T^{3} + 19454 T^{4} + 320 p T^{5} + 208 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 324 T^{2} - 3926 T^{3} + 41126 T^{4} - 3926 p T^{5} + 324 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 196 T^{2} + 2690 T^{3} + 18742 T^{4} + 2690 p T^{5} + 196 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 67 T^{2} + 32 T^{3} + 6240 T^{4} + 32 p T^{5} + 67 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 348 T^{2} + 4586 T^{3} + 51078 T^{4} + 4586 p T^{5} + 348 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.481334370762593863836516670310, −8.195822582563093419338897438413, −8.172824529198231833814855520098, −7.906494491978344271111571696771, −7.66660610259077591111994634778, −7.31937622994003106393735664024, −7.13525267508195215340102311921, −7.08830874552896084502705225365, −6.55467192115314912036789786229, −6.42977435960343359446140168654, −6.29960085958085693732772872556, −5.94610588804771616400788435859, −5.43442329951346623055725652593, −5.17345265331665152968490762031, −4.91064908322104063662949674689, −4.87743515483435024311432960468, −4.64370520967590416973420014453, −4.15831804220320238566162843221, −4.01526746204417431489995931593, −3.66294992069654980628268562429, −3.17707041902984177898901916665, −2.70587088047337827461922603862, −2.12071642513684236974011304291, −1.64831522406643035234155751022, −1.62177486373104229396777593335, 0, 0, 0, 0, 1.62177486373104229396777593335, 1.64831522406643035234155751022, 2.12071642513684236974011304291, 2.70587088047337827461922603862, 3.17707041902984177898901916665, 3.66294992069654980628268562429, 4.01526746204417431489995931593, 4.15831804220320238566162843221, 4.64370520967590416973420014453, 4.87743515483435024311432960468, 4.91064908322104063662949674689, 5.17345265331665152968490762031, 5.43442329951346623055725652593, 5.94610588804771616400788435859, 6.29960085958085693732772872556, 6.42977435960343359446140168654, 6.55467192115314912036789786229, 7.08830874552896084502705225365, 7.13525267508195215340102311921, 7.31937622994003106393735664024, 7.66660610259077591111994634778, 7.906494491978344271111571696771, 8.172824529198231833814855520098, 8.195822582563093419338897438413, 8.481334370762593863836516670310

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