Properties

Label 8-2475e4-1.1-c1e4-0-7
Degree $8$
Conductor $3.752\times 10^{13}$
Sign $1$
Analytic cond. $152548.$
Root an. cond. $4.44555$
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  + 2·4-s + 4·11-s + 3·16-s + 16·19-s − 8·29-s − 8·41-s + 8·44-s + 4·49-s − 16·59-s − 24·61-s + 12·64-s − 32·71-s + 32·76-s − 8·89-s − 8·101-s − 24·109-s − 16·116-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 16·164-s + ⋯
L(s)  = 1  + 4-s + 1.20·11-s + 3/4·16-s + 3.67·19-s − 1.48·29-s − 1.24·41-s + 1.20·44-s + 4/7·49-s − 2.08·59-s − 3.07·61-s + 3/2·64-s − 3.79·71-s + 3.67·76-s − 0.847·89-s − 0.796·101-s − 2.29·109-s − 1.48·116-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s − 1.24·164-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(152548.\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.476167857\)
\(L(\frac12)\) \(\approx\) \(6.476167857\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{4} \)
good2$D_4\times C_2$ \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
7$C_4\times C_2$ \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 - 12 T^{2} - 1834 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 84 T^{2} + 4310 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 52 T^{2} + 4246 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
61$D_{4}$ \( ( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 252 T^{2} + 30086 T^{4} - 252 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.38783582181425935419151937416, −6.08661421270061694517609354089, −5.84624012248208243183898944078, −5.73112847783807713090698179165, −5.59249373788849695434543582942, −5.45866781020379517862430150697, −5.19714322992353404529127091013, −4.81682428111886250463378931661, −4.62629272554037818486928924440, −4.46900875007296364615234843844, −4.33552617551814117429856230038, −3.88378118846950798331584440067, −3.68486096630303576990780475333, −3.44076808623735226929035088474, −3.20794363665649943270852398275, −3.08515366300691333099474903900, −2.99461009951280061993440136305, −2.53329294135711272127591792732, −2.44135873925448570708139591146, −1.65623609262533898635247304361, −1.62503927732425910002157252449, −1.52319634873200051154830910074, −1.39794245395925405540891681939, −0.72191048936972842072221422669, −0.41187736469541232322826991742, 0.41187736469541232322826991742, 0.72191048936972842072221422669, 1.39794245395925405540891681939, 1.52319634873200051154830910074, 1.62503927732425910002157252449, 1.65623609262533898635247304361, 2.44135873925448570708139591146, 2.53329294135711272127591792732, 2.99461009951280061993440136305, 3.08515366300691333099474903900, 3.20794363665649943270852398275, 3.44076808623735226929035088474, 3.68486096630303576990780475333, 3.88378118846950798331584440067, 4.33552617551814117429856230038, 4.46900875007296364615234843844, 4.62629272554037818486928924440, 4.81682428111886250463378931661, 5.19714322992353404529127091013, 5.45866781020379517862430150697, 5.59249373788849695434543582942, 5.73112847783807713090698179165, 5.84624012248208243183898944078, 6.08661421270061694517609354089, 6.38783582181425935419151937416

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