Properties

Label 8-60e4-1.1-c2e4-0-1
Degree $8$
Conductor $12960000$
Sign $1$
Analytic cond. $7.14406$
Root an. cond. $1.27862$
Motivic weight $2$
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  + 4·4-s + 6·9-s − 50·25-s + 144·29-s + 24·36-s + 72·41-s + 20·49-s − 296·61-s − 64·64-s + 27·81-s + 72·89-s − 200·100-s + 144·101-s + 104·109-s + 576·116-s + 268·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 288·164-s + 167-s + 28·169-s + ⋯
L(s)  = 1  + 4-s + 2/3·9-s − 2·25-s + 4.96·29-s + 2/3·36-s + 1.75·41-s + 0.408·49-s − 4.85·61-s − 64-s + 1/3·81-s + 0.808·89-s − 2·100-s + 1.42·101-s + 0.954·109-s + 4.96·116-s + 2.21·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 1.75·164-s + 0.00598·167-s + 0.165·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(7.14406\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{60} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 12960000,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.105063409\)
\(L(\frac12)\) \(\approx\) \(2.105063409\)
\(L(2)\) 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$C_2^2$ \( 1 - p^{2} T^{2} + p^{4} T^{4} \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
good7$C_2^2$ \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 134 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 478 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 530 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 36 T + p^{2} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 1874 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 178 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 3266 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
53$C_2^2$ \( ( 1 - 4942 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 5990 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 74 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 7250 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 718 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 9362 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 4370 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 5666 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 13634 T^{2} + p^{4} T^{4} )^{2} \)
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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

−11.15166645382640786618117830741, −10.55420215064691143691321790971, −10.24941218672404963073890935219, −10.17171427084914204247436510682, −10.15919050995604591110642248119, −9.277973850975125519896226242034, −9.244690731137069458376173257719, −8.878908673860368783240290797367, −8.436611060994473593840948569726, −7.992975152404441093536453226864, −7.68290578383106784099976878262, −7.60897122852066003840228852115, −7.02873970708748645772864447080, −6.74629279012048726898290484259, −6.18785581928845022646909641453, −6.12348328898362290817991877767, −5.94485614953169431329086413970, −4.95892377775252518668236308255, −4.63458114106254751109070966319, −4.47763910781306881374753849106, −3.82152555811029791515926039426, −3.04697893126555157947514251848, −2.73947392994343412870070891116, −2.06380075067946740643301729698, −1.16514509826015671606039577509, 1.16514509826015671606039577509, 2.06380075067946740643301729698, 2.73947392994343412870070891116, 3.04697893126555157947514251848, 3.82152555811029791515926039426, 4.47763910781306881374753849106, 4.63458114106254751109070966319, 4.95892377775252518668236308255, 5.94485614953169431329086413970, 6.12348328898362290817991877767, 6.18785581928845022646909641453, 6.74629279012048726898290484259, 7.02873970708748645772864447080, 7.60897122852066003840228852115, 7.68290578383106784099976878262, 7.992975152404441093536453226864, 8.436611060994473593840948569726, 8.878908673860368783240290797367, 9.244690731137069458376173257719, 9.277973850975125519896226242034, 10.15919050995604591110642248119, 10.17171427084914204247436510682, 10.24941218672404963073890935219, 10.55420215064691143691321790971, 11.15166645382640786618117830741

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