Properties

Label 4-825e2-1.1-c3e2-0-5
Degree $4$
Conductor $680625$
Sign $1$
Analytic cond. $2369.40$
Root an. cond. $6.97686$
Motivic weight $3$
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  + 15·4-s − 9·9-s + 22·11-s + 161·16-s + 120·19-s + 180·29-s − 16·31-s − 135·36-s + 844·41-s + 330·44-s + 10·49-s + 400·59-s + 264·61-s + 1.45e3·64-s + 1.52e3·71-s + 1.80e3·76-s + 1.10e3·79-s + 81·81-s − 1.14e3·89-s − 198·99-s + 3.40e3·101-s + 640·109-s + 2.70e3·116-s + 363·121-s − 240·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 15/8·4-s − 1/3·9-s + 0.603·11-s + 2.51·16-s + 1.44·19-s + 1.15·29-s − 0.0926·31-s − 5/8·36-s + 3.21·41-s + 1.13·44-s + 0.0291·49-s + 0.882·59-s + 0.554·61-s + 2.84·64-s + 2.54·71-s + 2.71·76-s + 1.56·79-s + 1/9·81-s − 1.35·89-s − 0.201·99-s + 3.35·101-s + 0.562·109-s + 2.16·116-s + 3/11·121-s − 0.173·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(680625\)    =    \(3^{2} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(2369.40\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 680625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.794247071\)
\(L(\frac12)\) \(\approx\) \(7.794247071\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p^{2} T^{2} \)
5 \( 1 \)
11$C_1$ \( ( 1 - p T )^{2} \)
good2$C_2^2$ \( 1 - 15 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 3370 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 4350 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 60 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 8790 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 90 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 96950 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 422 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 7450 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 48390 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 176650 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 200 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 132 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 471770 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 762 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 484270 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 550 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 1126150 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 570 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 1825150 T^{2} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.01120333756950394034449735719, −9.796455685942490626700695552492, −9.198672754388914411758559260146, −8.860100217433436023152778046512, −8.079264862538169184606742133672, −7.918501084000657947375197863043, −7.31119076985933211315344881999, −7.18190199638508235648026802132, −6.41323530984700875373023987972, −6.33485141451524277385433051572, −5.80459015168028023573784081393, −5.28928287962058318132939894842, −4.81827071383428850865880771738, −3.88735800724838438680904365490, −3.60151395813606525078454654794, −2.87264938533099546527297128886, −2.50121057571767823018062024288, −2.01719518082683411702678884092, −1.01986425254504192699302747427, −0.900957363883468340896770641342, 0.900957363883468340896770641342, 1.01986425254504192699302747427, 2.01719518082683411702678884092, 2.50121057571767823018062024288, 2.87264938533099546527297128886, 3.60151395813606525078454654794, 3.88735800724838438680904365490, 4.81827071383428850865880771738, 5.28928287962058318132939894842, 5.80459015168028023573784081393, 6.33485141451524277385433051572, 6.41323530984700875373023987972, 7.18190199638508235648026802132, 7.31119076985933211315344881999, 7.918501084000657947375197863043, 8.079264862538169184606742133672, 8.860100217433436023152778046512, 9.198672754388914411758559260146, 9.796455685942490626700695552492, 10.01120333756950394034449735719

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