Properties

Label 2-115920-1.1-c1-0-101
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s + 4·11-s − 2·13-s + 2·17-s − 4·19-s − 23-s + 25-s − 6·29-s + 8·31-s − 35-s − 2·37-s − 2·41-s + 4·47-s + 49-s − 6·53-s + 4·55-s − 8·59-s + 2·61-s − 2·65-s − 8·67-s + 16·71-s + 6·73-s − 4·77-s + 4·79-s − 12·83-s + 2·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.169·35-s − 0.328·37-s − 0.312·41-s + 0.583·47-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 − 0.977·67-s + 1.89·71-s + 0.702·73-s − 0.455·77-s + 0.450·79-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−13.85279910191999, −13.41218948359243, −12.83597396365107, −12.33803559532183, −12.02891548993969, −11.48543998797352, −10.85923771929736, −10.44947711923058, −9.720972402573761, −9.629268251743774, −8.952747006973749, −8.561964630625165, −7.850448761093443, −7.403594644872392, −6.689086963107328, −6.350385706407424, −5.960604743523784, −5.187476255684931, −4.720288997734602, −4.012902751181622, −3.601439313218506, −2.873928907636128, −2.227648708652547, −1.628824443820011, −0.9129832698701078, 0, 0.9129832698701078, 1.628824443820011, 2.227648708652547, 2.873928907636128, 3.601439313218506, 4.012902751181622, 4.720288997734602, 5.187476255684931, 5.960604743523784, 6.350385706407424, 6.689086963107328, 7.403594644872392, 7.850448761093443, 8.561964630625165, 8.952747006973749, 9.629268251743774, 9.720972402573761, 10.44947711923058, 10.85923771929736, 11.48543998797352, 12.02891548993969, 12.33803559532183, 12.83597396365107, 13.41218948359243, 13.85279910191999

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