Properties

Label 2-115920-1.1-c1-0-134
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 2·11-s − 4·13-s − 2·17-s − 8·19-s + 23-s + 25-s − 6·29-s − 35-s + 8·37-s + 10·41-s − 10·43-s − 8·47-s + 49-s − 12·53-s + 2·55-s + 6·59-s + 10·61-s + 4·65-s − 10·67-s − 10·73-s − 2·77-s + 4·79-s − 16·83-s + 2·85-s − 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.603·11-s − 1.10·13-s − 0.485·17-s − 1.83·19-s + 0.208·23-s + 1/5·25-s − 1.11·29-s − 0.169·35-s + 1.31·37-s + 1.56·41-s − 1.52·43-s − 1.16·47-s + 1/7·49-s − 1.64·53-s + 0.269·55-s + 0.781·59-s + 1.28·61-s + 0.496·65-s − 1.22·67-s − 1.17·73-s − 0.227·77-s + 0.450·79-s − 1.75·83-s + 0.216·85-s − 0.635·89-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: $\chi_{115920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ 1)\)

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$$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 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.28689337925466, −13.38092286842986, −13.09587305072496, −12.63449026272301, −12.32166774343044, −11.40128555666134, −11.26941688781959, −10.87016751382831, −10.02572216794166, −9.893126221065674, −9.124449291056005, −8.610432523891066, −8.143408859898958, −7.678327882630430, −7.214100518462766, −6.615005501592327, −6.109930031878970, −5.434208234717234, −4.868010164461468, −4.380509423373966, −4.014541716328848, −3.109728569255845, −2.532648393839875, −2.055324390674175, −1.287773265983660, 0, 0, 1.287773265983660, 2.055324390674175, 2.532648393839875, 3.109728569255845, 4.014541716328848, 4.380509423373966, 4.868010164461468, 5.434208234717234, 6.109930031878970, 6.615005501592327, 7.214100518462766, 7.678327882630430, 8.143408859898958, 8.610432523891066, 9.124449291056005, 9.893126221065674, 10.02572216794166, 10.87016751382831, 11.26941688781959, 11.40128555666134, 12.32166774343044, 12.63449026272301, 13.09587305072496, 13.38092286842986, 14.28689337925466

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