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Properties

Label 2-115920-1.1-c1-0-15
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 + 3·11-s − 4·13-s + 6·17-s + 19-s − 23-s + 25-s − 6·29-s − 8·31-s − 35-s − 10·37-s + 3·41-s + 10·43-s − 3·47-s + 49-s + 9·53-s + 3·55-s − 3·59-s + 11·61-s − 4·65-s + 10·67-s + 2·73-s − 3·77-s + 10·79-s + 6·85-s + 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.904·11-s − 1.10·13-s + 1.45·17-s + 0.229·19-s − 0.208·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.169·35-s − 1.64·37-s + 0.468·41-s + 1.52·43-s − 0.437·47-s + 1/7·49-s + 1.23·53-s + 0.404·55-s − 0.390·59-s + 1.40·61-s − 0.496·65-s + 1.22·67-s + 0.234·73-s − 0.341·77-s + 1.12·79-s + 0.650·85-s + 0.419·91-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

\(L(1)\) \(\approx\) \(2.419711667\)
\(L(\frac12)\) \(\approx\) \(2.419711667\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.58579030140752, −13.13205303944818, −12.43765218343144, −12.28581188664090, −11.84001538941418, −11.08160196671823, −10.73101329994277, −10.00861975266870, −9.695387638495655, −9.294579658364710, −8.856736345317616, −8.107848210366255, −7.564256237492964, −7.051650536670119, −6.782706894766232, −5.847098983649417, −5.581832880937509, −5.143575514627820, −4.341753385944655, −3.565205059596014, −3.484220335112955, −2.449969392101644, −2.009499268004593, −1.249580813575632, −0.4939730060703514, 0.4939730060703514, 1.249580813575632, 2.009499268004593, 2.449969392101644, 3.484220335112955, 3.565205059596014, 4.341753385944655, 5.143575514627820, 5.581832880937509, 5.847098983649417, 6.782706894766232, 7.051650536670119, 7.564256237492964, 8.107848210366255, 8.856736345317616, 9.294579658364710, 9.695387638495655, 10.00861975266870, 10.73101329994277, 11.08160196671823, 11.84001538941418, 12.28581188664090, 12.43765218343144, 13.13205303944818, 13.58579030140752

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