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Properties

Label 2-115920-1.1-c1-0-2
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 − 4·11-s − 2·13-s + 6·17-s + 23-s + 25-s + 2·29-s − 8·31-s + 35-s + 2·37-s − 2·41-s + 4·43-s − 8·47-s + 49-s − 6·53-s + 4·55-s + 8·59-s − 2·61-s + 2·65-s − 4·67-s − 12·71-s − 14·73-s + 4·77-s + 12·79-s − 12·83-s − 6·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.20·11-s − 0.554·13-s + 1.45·17-s + 0.208·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.169·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.539·55-s + 1.04·59-s − 0.256·61-s + 0.248·65-s − 0.488·67-s − 1.42·71-s − 1.63·73-s + 0.455·77-s + 1.35·79-s − 1.31·83-s − 0.650·85-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

\(L(1)\) \(\approx\) \(0.8609665765\)
\(L(\frac12)\) \(\approx\) \(0.8609665765\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
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

−13.42391612969789, −13.06624132298149, −12.67956406309580, −12.19238615906072, −11.71187528415797, −11.21010696822762, −10.57542783085386, −10.21011765970900, −9.816810610844286, −9.173671448617525, −8.709073670229225, −7.926780659567486, −7.754299604861011, −7.232521895655479, −6.714041664688104, −5.831960883495211, −5.631693400077899, −4.896696583518921, −4.534108942975992, −3.636406589670235, −3.221877808563663, −2.727267672368486, −1.981518738527322, −1.178055760922362, −0.2992155287417937, 0.2992155287417937, 1.178055760922362, 1.981518738527322, 2.727267672368486, 3.221877808563663, 3.636406589670235, 4.534108942975992, 4.896696583518921, 5.631693400077899, 5.831960883495211, 6.714041664688104, 7.232521895655479, 7.754299604861011, 7.926780659567486, 8.709073670229225, 9.173671448617525, 9.816810610844286, 10.21011765970900, 10.57542783085386, 11.21010696822762, 11.71187528415797, 12.19238615906072, 12.67956406309580, 13.06624132298149, 13.42391612969789

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