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Properties

Label 8-574e4-1.1-c1e4-0-7
Degree $8$
Conductor $108554434576$
Sign $1$
Analytic cond. $441.322$
Root an. cond. $2.14089$
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 − 3-s + 10·4-s + 3·5-s + 4·6-s − 4·7-s − 20·8-s − 9-s − 12·10-s + 4·11-s − 10·12-s − 6·13-s + 16·14-s − 3·15-s + 35·16-s − 17-s + 4·18-s + 2·19-s + 30·20-s + 4·21-s − 16·22-s + 6·23-s + 20·24-s + 25-s + 24·26-s − 2·27-s − 40·28-s + ⋯
L(s)  = 1  − 2.82·2-s − 0.577·3-s + 5·4-s + 1.34·5-s + 1.63·6-s − 1.51·7-s − 7.07·8-s − 1/3·9-s − 3.79·10-s + 1.20·11-s − 2.88·12-s − 1.66·13-s + 4.27·14-s − 0.774·15-s + 35/4·16-s − 0.242·17-s + 0.942·18-s + 0.458·19-s + 6.70·20-s + 0.872·21-s − 3.41·22-s + 1.25·23-s + 4.08·24-s + 1/5·25-s + 4.70·26-s − 0.384·27-s − 7.55·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 7^{4} \cdot 41^{4}\)
Sign: $1$
Analytic conductor: \(441.322\)
Root analytic conductor: \(2.14089\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 7^{4} \cdot 41^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6133349726\)
\(L(\frac12)\) \(\approx\) \(0.6133349726\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
7$C_1$ \( ( 1 + T )^{4} \)
41$C_1$ \( ( 1 - T )^{4} \)
good3$C_2^3:S_4$ \( 1 + T + 2 T^{2} + 5 T^{3} + 2 T^{4} + 5 p T^{5} + 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 3 T + 8 T^{2} - p T^{3} + 2 T^{4} - p^{2} T^{5} + 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 4 T + 36 T^{2} - 104 T^{3} + 534 T^{4} - 104 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 6 T + 44 T^{2} + 170 T^{3} + 774 T^{4} + 170 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + T + 10 T^{2} + 23 T^{3} - 86 T^{4} + 23 p T^{5} + 10 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 2 T + 56 T^{2} - 106 T^{3} + 1438 T^{4} - 106 p T^{5} + 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 6 T + 84 T^{2} - 350 T^{3} + 2774 T^{4} - 350 p T^{5} + 84 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 17 T + 178 T^{2} - 1231 T^{3} + 7294 T^{4} - 1231 p T^{5} + 178 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 5 T + 68 T^{2} - 289 T^{3} + 2326 T^{4} - 289 p T^{5} + 68 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 6 T + 120 T^{2} + 546 T^{3} + 6174 T^{4} + 546 p T^{5} + 120 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 13 T + 220 T^{2} + 1709 T^{3} + 15190 T^{4} + 1709 p T^{5} + 220 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 6 T + 108 T^{2} - 190 T^{3} + 4550 T^{4} - 190 p T^{5} + 108 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 29 T + 362 T^{2} - 2619 T^{3} + 16622 T^{4} - 2619 p T^{5} + 362 p^{2} T^{6} - 29 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 16 T + 264 T^{2} - 2444 T^{3} + 23526 T^{4} - 2444 p T^{5} + 264 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 21 T + 276 T^{2} - 2979 T^{3} + 26922 T^{4} - 2979 p T^{5} + 276 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 4 T + 172 T^{2} - 384 T^{3} + 14822 T^{4} - 384 p T^{5} + 172 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 17 T + 84 T^{2} + 1019 T^{3} - 16170 T^{4} + 1019 p T^{5} + 84 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 12 T + 116 T^{2} - 276 T^{3} + 3382 T^{4} - 276 p T^{5} + 116 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 27 T + 548 T^{2} + 7039 T^{3} + 74358 T^{4} + 7039 p T^{5} + 548 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 18 T + 412 T^{2} + 4590 T^{3} + 54630 T^{4} + 4590 p T^{5} + 412 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 37 T + 814 T^{2} - 11891 T^{3} + 130434 T^{4} - 11891 p T^{5} + 814 p^{2} T^{6} - 37 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 3 T + 290 T^{2} + 621 T^{3} + 37786 T^{4} + 621 p T^{5} + 290 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.893023572216397360725236312558, −7.35629933919345843844553158803, −7.16569406458843020283252910507, −7.06274955828220202435048597931, −6.86554318761368352250794253374, −6.60084233518333489883423935489, −6.54520429416786073301734027519, −6.27329991151756514975737190268, −6.11645933976654703196940785373, −5.56573970648120326831243937314, −5.37130606536787360280416168736, −5.27721847804979497009182041416, −5.12676978873747803931935904066, −4.52502161207761104859134602259, −4.04113978157380851187750642065, −3.67814377867018393193813802700, −3.62859318308238588657517587892, −2.85321761632502117640457185816, −2.75706114457503498399189516382, −2.39820279324503093475235230142, −2.39650147332108740401764990499, −1.84049760559799226407337434598, −1.23486423036038882779714156875, −0.73367515690551728798983658049, −0.63967045228762778715899693228, 0.63967045228762778715899693228, 0.73367515690551728798983658049, 1.23486423036038882779714156875, 1.84049760559799226407337434598, 2.39650147332108740401764990499, 2.39820279324503093475235230142, 2.75706114457503498399189516382, 2.85321761632502117640457185816, 3.62859318308238588657517587892, 3.67814377867018393193813802700, 4.04113978157380851187750642065, 4.52502161207761104859134602259, 5.12676978873747803931935904066, 5.27721847804979497009182041416, 5.37130606536787360280416168736, 5.56573970648120326831243937314, 6.11645933976654703196940785373, 6.27329991151756514975737190268, 6.54520429416786073301734027519, 6.60084233518333489883423935489, 6.86554318761368352250794253374, 7.06274955828220202435048597931, 7.16569406458843020283252910507, 7.35629933919345843844553158803, 7.893023572216397360725236312558

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