Properties

Label 2-115920-1.1-c1-0-125
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 + 6·13-s − 2·17-s + 23-s + 25-s − 6·29-s − 35-s − 6·37-s + 6·41-s − 12·43-s + 8·47-s + 49-s + 10·53-s − 4·55-s − 2·61-s − 6·65-s − 4·67-s − 12·71-s − 14·73-s + 4·77-s + 4·79-s + 4·83-s + 2·85-s − 10·89-s + 6·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 1.20·11-s + 1.66·13-s − 0.485·17-s + 0.208·23-s + 1/5·25-s − 1.11·29-s − 0.169·35-s − 0.986·37-s + 0.937·41-s − 1.82·43-s + 1.16·47-s + 1/7·49-s + 1.37·53-s − 0.539·55-s − 0.256·61-s − 0.744·65-s − 0.488·67-s − 1.42·71-s − 1.63·73-s + 0.455·77-s + 0.450·79-s + 0.439·83-s + 0.216·85-s − 1.05·89-s + 0.628·91-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: $\chi_{115920} (1, \cdot )$
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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.73473668807667, −13.36246527689448, −13.06026569580614, −12.17822187083502, −11.91574841718411, −11.38572537378254, −11.06137569201624, −10.52123477784291, −10.05241228711042, −9.137804959478353, −8.924842201731204, −8.599275774187837, −7.938311814155838, −7.337807623415972, −6.884130704246833, −6.338164473989606, −5.819867097025639, −5.320304794655438, −4.467956159017976, −4.084479165978531, −3.607568121806128, −3.087941402602244, −2.128624723032323, −1.484814128150090, −1.018482381449417, 0, 1.018482381449417, 1.484814128150090, 2.128624723032323, 3.087941402602244, 3.607568121806128, 4.084479165978531, 4.467956159017976, 5.320304794655438, 5.819867097025639, 6.338164473989606, 6.884130704246833, 7.337807623415972, 7.938311814155838, 8.599275774187837, 8.924842201731204, 9.137804959478353, 10.05241228711042, 10.52123477784291, 11.06137569201624, 11.38572537378254, 11.91574841718411, 12.17822187083502, 13.06026569580614, 13.36246527689448, 13.73473668807667

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