Properties

Label 8-585e4-1.1-c1e4-0-8
Degree $8$
Conductor $117117950625$
Sign $1$
Analytic cond. $476.136$
Root an. cond. $2.16130$
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  − 8·16-s − 2·25-s − 16·43-s + 2·49-s + 52·61-s + 52·79-s + 40·103-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 2·16-s − 2/5·25-s − 2.43·43-s + 2/7·49-s + 6.65·61-s + 5.85·79-s + 3.94·103-s + 2.36·121-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 + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(476.136\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{585} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.353841925\)
\(L(\frac12)\) \(\approx\) \(2.353841925\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \)
7$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 21 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 33 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 39 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 11 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{4} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 117 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 169 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 181 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.81293144968649727380666148945, −7.35449638481312468048724572557, −7.03708597777763116281173647207, −7.03152230156419155182142912956, −6.94391595049713855404006931951, −6.48528091341689320842539822025, −6.33746068979094599465136030378, −6.13476236443244883773583525109, −5.87180431971600677317119580184, −5.33320711136710335634204686467, −5.15547989339809802879574550380, −5.12302110017609101935014833741, −4.79420707122433485941673250125, −4.56971035817774309661104182808, −4.12506382401174225492552797058, −3.76840870973028695816046339357, −3.75120582976975299775343947977, −3.38007843452385266801304455355, −3.06659558239422310515133068402, −2.50548538284112622547594275756, −2.17691516096619175552700088884, −2.03689542962607304281943648936, −1.85551754050527559636569970945, −0.70107886146530611510706438638, −0.69912277573216465176782182821, 0.69912277573216465176782182821, 0.70107886146530611510706438638, 1.85551754050527559636569970945, 2.03689542962607304281943648936, 2.17691516096619175552700088884, 2.50548538284112622547594275756, 3.06659558239422310515133068402, 3.38007843452385266801304455355, 3.75120582976975299775343947977, 3.76840870973028695816046339357, 4.12506382401174225492552797058, 4.56971035817774309661104182808, 4.79420707122433485941673250125, 5.12302110017609101935014833741, 5.15547989339809802879574550380, 5.33320711136710335634204686467, 5.87180431971600677317119580184, 6.13476236443244883773583525109, 6.33746068979094599465136030378, 6.48528091341689320842539822025, 6.94391595049713855404006931951, 7.03152230156419155182142912956, 7.03708597777763116281173647207, 7.35449638481312468048724572557, 7.81293144968649727380666148945

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