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Properties

Label 2-115920-1.1-c1-0-29
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 + 5·19-s − 23-s + 25-s + 4·29-s + 8·31-s − 35-s + 4·37-s + 9·41-s + 6·43-s − 47-s + 49-s + 11·53-s − 3·55-s + 9·59-s + 61-s − 4·65-s + 10·67-s − 12·71-s + 3·77-s − 4·79-s + 6·83-s − 12·89-s + 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.904·11-s − 1.10·13-s + 1.14·19-s − 0.208·23-s + 1/5·25-s + 0.742·29-s + 1.43·31-s − 0.169·35-s + 0.657·37-s + 1.40·41-s + 0.914·43-s − 0.145·47-s + 1/7·49-s + 1.51·53-s − 0.404·55-s + 1.17·59-s + 0.128·61-s − 0.496·65-s + 1.22·67-s − 1.42·71-s + 0.341·77-s − 0.450·79-s + 0.658·83-s − 1.27·89-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

\(L(1)\) \(\approx\) \(2.636492990\)
\(L(\frac12)\) \(\approx\) \(2.636492990\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−13.71476199799347, −13.09873644639132, −12.59144859408363, −12.31645282748109, −11.56955648133142, −11.35024167428723, −10.45359733655560, −10.10625739220312, −9.858572812864308, −9.272751425427633, −8.732299480930511, −8.082924910073053, −7.631794932740118, −7.161110906566416, −6.651450153669117, −5.895968438600821, −5.609138724809763, −4.959688612552763, −4.496030542743105, −3.847821324870152, −2.930314984799333, −2.651310769689668, −2.168459494078295, −1.063820894644762, −0.5668438020501568, 0.5668438020501568, 1.063820894644762, 2.168459494078295, 2.651310769689668, 2.930314984799333, 3.847821324870152, 4.496030542743105, 4.959688612552763, 5.609138724809763, 5.895968438600821, 6.651450153669117, 7.161110906566416, 7.631794932740118, 8.082924910073053, 8.732299480930511, 9.272751425427633, 9.858572812864308, 10.10625739220312, 10.45359733655560, 11.35024167428723, 11.56955648133142, 12.31645282748109, 12.59144859408363, 13.09873644639132, 13.71476199799347

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