Properties

Label 4-7488e2-1.1-c1e2-0-4
Degree $4$
Conductor $56070144$
Sign $1$
Analytic cond. $3575.08$
Root an. cond. $7.73252$
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  − 2·5-s + 6·7-s + 2·13-s − 2·19-s − 4·23-s − 2·25-s − 8·29-s + 14·31-s − 12·35-s + 14·41-s − 4·43-s − 8·47-s + 18·49-s − 8·53-s + 4·59-s − 16·61-s − 4·65-s + 2·67-s − 20·71-s + 12·83-s + 18·89-s + 12·91-s + 4·95-s + 24·97-s + 20·101-s + 4·103-s + 4·107-s + ⋯
L(s)  = 1  − 0.894·5-s + 2.26·7-s + 0.554·13-s − 0.458·19-s − 0.834·23-s − 2/5·25-s − 1.48·29-s + 2.51·31-s − 2.02·35-s + 2.18·41-s − 0.609·43-s − 1.16·47-s + 18/7·49-s − 1.09·53-s + 0.520·59-s − 2.04·61-s − 0.496·65-s + 0.244·67-s − 2.37·71-s + 1.31·83-s + 1.90·89-s + 1.25·91-s + 0.410·95-s + 2.43·97-s + 1.99·101-s + 0.394·103-s + 0.386·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(56070144\)    =    \(2^{12} \cdot 3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3575.08\)
Root analytic conductor: \(7.73252\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 56070144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.865296771\)
\(L(\frac12)\) \(\approx\) \(2.865296771\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$C_4$ \( 1 - 14 T + 106 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_4$ \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 24 T + 318 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
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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

−7.926489408460554011770534644484, −7.85459387459277641648216490126, −7.54049714042846935094025442442, −7.33080727485402191519169657598, −6.51714408628138160579642281118, −6.36966337736594870299495148920, −5.87780850815003785958240140050, −5.79470334361140405204788218373, −4.93270581481378579724642753458, −4.90870218899717124800131522706, −4.44867801724661209654420009572, −4.35857028531152284536671161288, −3.73350579332389387793333539906, −3.52965832653256396852235581183, −2.88802539733523814497018709873, −2.41966596860460804452903040820, −1.77520191729155550262632348087, −1.73299118745867855814196444418, −1.04057333024007474020774710321, −0.44470686000457059600434419072, 0.44470686000457059600434419072, 1.04057333024007474020774710321, 1.73299118745867855814196444418, 1.77520191729155550262632348087, 2.41966596860460804452903040820, 2.88802539733523814497018709873, 3.52965832653256396852235581183, 3.73350579332389387793333539906, 4.35857028531152284536671161288, 4.44867801724661209654420009572, 4.90870218899717124800131522706, 4.93270581481378579724642753458, 5.79470334361140405204788218373, 5.87780850815003785958240140050, 6.36966337736594870299495148920, 6.51714408628138160579642281118, 7.33080727485402191519169657598, 7.54049714042846935094025442442, 7.85459387459277641648216490126, 7.926489408460554011770534644484

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