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Properties

Label 2-115920-1.1-c1-0-3
Degree $2$
Conductor $115920$
Sign $1$
Analytic cond. $925.625$
Root an. cond. $30.4240$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 2·11-s − 4·13-s + 2·17-s + 4·19-s + 23-s + 25-s + 2·29-s + 4·31-s + 35-s − 8·37-s + 2·41-s − 10·43-s − 12·47-s + 49-s + 4·53-s + 2·55-s + 10·59-s − 14·61-s + 4·65-s − 10·67-s − 16·71-s + 6·73-s + 2·77-s + 4·79-s + 12·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.603·11-s − 1.10·13-s + 0.485·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.169·35-s − 1.31·37-s + 0.312·41-s − 1.52·43-s − 1.75·47-s + 1/7·49-s + 0.549·53-s + 0.269·55-s + 1.30·59-s − 1.79·61-s + 0.496·65-s − 1.22·67-s − 1.89·71-s + 0.702·73-s + 0.227·77-s + 0.450·79-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(925.625\)
Root analytic conductor: \(30.4240\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8937131158\)
\(L(\frac12)\) \(\approx\) \(0.8937131158\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.55584292553265, −13.16674554404008, −12.52781564929589, −12.14963205572115, −11.71860094894677, −11.35424237496835, −10.48890882395958, −10.15659181114763, −9.878457735194271, −9.169628806583004, −8.682408749011113, −8.107422594511947, −7.545640457640715, −7.288957349720458, −6.635389125973170, −6.103875525430352, −5.393278425983999, −4.843341712018568, −4.660416234496521, −3.580893970289834, −3.260472056020454, −2.721346348402753, −1.971392539489654, −1.185609867119368, −0.3041613752160189, 0.3041613752160189, 1.185609867119368, 1.971392539489654, 2.721346348402753, 3.260472056020454, 3.580893970289834, 4.660416234496521, 4.843341712018568, 5.393278425983999, 6.103875525430352, 6.635389125973170, 7.288957349720458, 7.545640457640715, 8.107422594511947, 8.682408749011113, 9.169628806583004, 9.878457735194271, 10.15659181114763, 10.48890882395958, 11.35424237496835, 11.71860094894677, 12.14963205572115, 12.52781564929589, 13.16674554404008, 13.55584292553265

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