Properties

Label 8-1875e4-1.1-c1e4-0-3
Degree $8$
Conductor $1.236\times 10^{13}$
Sign $1$
Analytic cond. $50247.3$
Root an. cond. $3.86936$
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  − 2·2-s − 4·3-s + 2·4-s + 8·6-s − 2·7-s − 5·8-s + 10·9-s − 7·11-s − 8·12-s − 13-s + 4·14-s + 5·16-s − 2·17-s − 20·18-s + 5·19-s + 8·21-s + 14·22-s − 23-s + 20·24-s + 2·26-s − 20·27-s − 4·28-s − 20·29-s + 23·31-s + 2·32-s + 28·33-s + 4·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 4-s + 3.26·6-s − 0.755·7-s − 1.76·8-s + 10/3·9-s − 2.11·11-s − 2.30·12-s − 0.277·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s − 4.71·18-s + 1.14·19-s + 1.74·21-s + 2.98·22-s − 0.208·23-s + 4.08·24-s + 0.392·26-s − 3.84·27-s − 0.755·28-s − 3.71·29-s + 4.13·31-s + 0.353·32-s + 4.87·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(50247.3\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
good2$C_2^2:C_4$ \( 1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 12 T^{2} + 15 T^{3} + 61 T^{4} + 15 p T^{5} + 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 38 T^{2} + 139 T^{3} + 485 T^{4} + 139 p T^{5} + 38 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + T + 38 T^{2} + 15 T^{3} + 641 T^{4} + 15 p T^{5} + 38 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 62 T^{2} + 95 T^{3} + 1541 T^{4} + 95 p T^{5} + 62 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 41 T^{2} - 210 T^{3} + 1111 T^{4} - 210 p T^{5} + 41 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + T + 68 T^{2} + 5 p T^{3} + 2051 T^{4} + 5 p^{2} T^{5} + 68 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 256 T^{2} + 2145 T^{3} + 13571 T^{4} + 2145 p T^{5} + 256 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 23 T + 308 T^{2} - 2751 T^{3} + 17885 T^{4} - 2751 p T^{5} + 308 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 102 T^{2} + 175 T^{3} + 4811 T^{4} + 175 p T^{5} + 102 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 233 T^{2} + 2040 T^{3} + 16241 T^{4} + 2040 p T^{5} + 233 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 152 T^{2} + 245 T^{3} + 10181 T^{4} + 245 p T^{5} + 152 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 178 T^{2} - 585 T^{3} + 13511 T^{4} - 585 p T^{5} + 178 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 191 T^{2} + 1440 T^{3} + 11931 T^{4} + 1440 p T^{5} + 191 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 188 T^{2} + 444 T^{3} + 15485 T^{4} + 444 p T^{5} + 188 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 237 T^{2} + 390 T^{3} + 22951 T^{4} + 390 p T^{5} + 237 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 68 T^{2} - 811 T^{3} + 485 T^{4} - 811 p T^{5} + 68 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 243 T^{2} + 2010 T^{3} + 19951 T^{4} + 2010 p T^{5} + 243 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 35 T + 636 T^{2} - 7945 T^{3} + 78161 T^{4} - 7945 p T^{5} + 636 p^{2} T^{6} - 35 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 278 T^{2} + 2680 T^{3} + 31391 T^{4} + 2680 p T^{5} + 278 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 35 T + 776 T^{2} + 11375 T^{3} + 125591 T^{4} + 11375 p T^{5} + 776 p^{2} T^{6} + 35 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 302 T^{2} + 2760 T^{3} + 41871 T^{4} + 2760 p T^{5} + 302 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.79041910701099933218731601384, −6.75438441569248784658587142371, −6.72652405724055332287572756453, −6.36568892921692659120990516883, −6.09231999768110555365368894600, −5.89268101871923316925531281612, −5.87654526128136849560557566857, −5.67618882811943976474042210756, −5.26062967649163010045062329301, −5.10450689436898656332787697746, −4.94794900323656056423497291313, −4.86699712291511602921089773764, −4.57254999025850795464794371669, −4.26144907607048968538838390580, −4.01949662092914436300142946813, −3.53096067251791224582227547948, −3.38472164127444139805437488134, −3.19448320551249347297506611555, −2.85620781414208858062194040520, −2.59848196524728356400407954996, −2.38139094590099704251695070576, −1.99038850647125296398043019466, −1.61262689078604032373439999062, −1.19850140155590226581111780388, −1.08163407947663781807992829612, 0, 0, 0, 0, 1.08163407947663781807992829612, 1.19850140155590226581111780388, 1.61262689078604032373439999062, 1.99038850647125296398043019466, 2.38139094590099704251695070576, 2.59848196524728356400407954996, 2.85620781414208858062194040520, 3.19448320551249347297506611555, 3.38472164127444139805437488134, 3.53096067251791224582227547948, 4.01949662092914436300142946813, 4.26144907607048968538838390580, 4.57254999025850795464794371669, 4.86699712291511602921089773764, 4.94794900323656056423497291313, 5.10450689436898656332787697746, 5.26062967649163010045062329301, 5.67618882811943976474042210756, 5.87654526128136849560557566857, 5.89268101871923316925531281612, 6.09231999768110555365368894600, 6.36568892921692659120990516883, 6.72652405724055332287572756453, 6.75438441569248784658587142371, 6.79041910701099933218731601384

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