Properties

Label 4-9280e2-1.1-c1e2-0-8
Degree $4$
Conductor $86118400$
Sign $1$
Analytic cond. $5490.98$
Root an. cond. $8.60820$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 2·5-s − 4·7-s + 6·9-s + 4·11-s + 4·13-s − 8·15-s + 4·19-s − 16·21-s − 12·23-s + 3·25-s − 4·27-s − 2·29-s − 4·31-s + 16·33-s + 8·35-s + 16·39-s − 12·41-s + 12·43-s − 12·45-s − 12·47-s + 6·49-s − 4·53-s − 8·55-s + 16·57-s − 4·61-s − 24·63-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.894·5-s − 1.51·7-s + 2·9-s + 1.20·11-s + 1.10·13-s − 2.06·15-s + 0.917·19-s − 3.49·21-s − 2.50·23-s + 3/5·25-s − 0.769·27-s − 0.371·29-s − 0.718·31-s + 2.78·33-s + 1.35·35-s + 2.56·39-s − 1.87·41-s + 1.82·43-s − 1.78·45-s − 1.75·47-s + 6/7·49-s − 0.549·53-s − 1.07·55-s + 2.11·57-s − 0.512·61-s − 3.02·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86118400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86118400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86118400\)    =    \(2^{12} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(5490.98\)
Root analytic conductor: \(8.60820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 86118400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
31$C_4$ \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−7.55004726962221854490502348695, −7.52935841445792719656894942042, −6.76124291237132002564778953053, −6.75742020811871498910511525371, −6.16141453895434646025068463434, −6.08450676513349375824635761418, −5.44960961189029285722321006335, −5.26945393637055222510735741281, −4.22725262982081824486802949859, −4.19361117178363989360710914088, −3.81022237483376966484214778935, −3.62207903602386356609748815468, −3.23863653176883036874670361482, −3.00879550758426293486779513315, −2.63927096862702627230244967160, −2.02671051010642108883898894786, −1.61803862594263952614305147078, −1.23420808500980920671183249328, 0, 0, 1.23420808500980920671183249328, 1.61803862594263952614305147078, 2.02671051010642108883898894786, 2.63927096862702627230244967160, 3.00879550758426293486779513315, 3.23863653176883036874670361482, 3.62207903602386356609748815468, 3.81022237483376966484214778935, 4.19361117178363989360710914088, 4.22725262982081824486802949859, 5.26945393637055222510735741281, 5.44960961189029285722321006335, 6.08450676513349375824635761418, 6.16141453895434646025068463434, 6.75742020811871498910511525371, 6.76124291237132002564778953053, 7.52935841445792719656894942042, 7.55004726962221854490502348695

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