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Properties

Label 2-115920-1.1-c1-0-135
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 − 4·11-s − 2·13-s − 6·17-s + 4·19-s − 23-s + 25-s − 2·29-s − 4·31-s − 35-s − 6·37-s − 10·41-s − 4·43-s − 12·47-s + 49-s + 6·53-s − 4·55-s − 4·59-s − 14·61-s − 2·65-s + 12·67-s − 12·71-s + 10·73-s + 4·77-s + 8·79-s − 4·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 1.20·11-s − 0.554·13-s − 1.45·17-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 − 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s − 0.539·55-s − 0.520·59-s − 1.79·61-s − 0.248·65-s + 1.46·67-s − 1.42·71-s + 1.17·73-s + 0.455·77-s + 0.900·79-s − 0.439·83-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: $\chi_{115920} (1, \cdot )$
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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.77568198323698, −13.55037706359698, −13.33841661014723, −12.55728004743000, −12.34329481809824, −11.61943131928276, −11.13507019732347, −10.65145225686049, −10.16640038103989, −9.702887305389552, −9.302319695167137, −8.649081858465116, −8.240771399321629, −7.588395981767783, −7.079876626576408, −6.663545868006133, −6.063117008192077, −5.402240456681598, −4.994521500784240, −4.618003940135024, −3.611780652778191, −3.257990653593156, −2.494032357782521, −2.042040376783254, −1.363217932079803, 0, 0, 1.363217932079803, 2.042040376783254, 2.494032357782521, 3.257990653593156, 3.611780652778191, 4.618003940135024, 4.994521500784240, 5.402240456681598, 6.063117008192077, 6.663545868006133, 7.079876626576408, 7.588395981767783, 8.240771399321629, 8.649081858465116, 9.302319695167137, 9.702887305389552, 10.16640038103989, 10.65145225686049, 11.13507019732347, 11.61943131928276, 12.34329481809824, 12.55728004743000, 13.33841661014723, 13.55037706359698, 13.77568198323698

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