Properties

Label 2-115920-1.1-c1-0-0
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 − 5·11-s + 4·13-s − 2·17-s − 5·19-s − 23-s + 25-s + 2·29-s − 8·31-s − 35-s − 6·37-s + 41-s − 10·43-s − 7·47-s + 49-s − 53-s + 5·55-s − 7·59-s + 11·61-s − 4·65-s − 2·67-s − 10·73-s − 5·77-s − 6·79-s + 12·83-s + 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 1.50·11-s + 1.10·13-s − 0.485·17-s − 1.14·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.169·35-s − 0.986·37-s + 0.156·41-s − 1.52·43-s − 1.02·47-s + 1/7·49-s − 0.137·53-s + 0.674·55-s − 0.911·59-s + 1.40·61-s − 0.496·65-s − 0.244·67-s − 1.17·73-s − 0.569·77-s − 0.675·79-s + 1.31·83-s + 0.216·85-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: $\chi_{115920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2776949702\)
\(L(\frac12)\) \(\approx\) \(0.2776949702\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.43847900301034, −13.14843775750308, −12.74826208341030, −12.19051784547822, −11.54545844655945, −11.13556895685459, −10.69556478839108, −10.37840548776786, −9.804628611229756, −8.941907701858072, −8.664581326518719, −8.121124369649531, −7.845781578223070, −7.091126809753317, −6.665316354574243, −6.035108617976650, −5.457334864438603, −4.942437887145562, −4.452516455436933, −3.735039638867588, −3.324578603683659, −2.544194929805380, −1.939913934509176, −1.331210215377965, −0.1578148459393393, 0.1578148459393393, 1.331210215377965, 1.939913934509176, 2.544194929805380, 3.324578603683659, 3.735039638867588, 4.452516455436933, 4.942437887145562, 5.457334864438603, 6.035108617976650, 6.665316354574243, 7.091126809753317, 7.845781578223070, 8.121124369649531, 8.664581326518719, 8.941907701858072, 9.804628611229756, 10.37840548776786, 10.69556478839108, 11.13556895685459, 11.54545844655945, 12.19051784547822, 12.74826208341030, 13.14843775750308, 13.43847900301034

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