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Properties

Label 2-115920-1.1-c1-0-116
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 + 6·11-s − 4·13-s − 2·17-s + 8·19-s − 23-s + 25-s − 2·29-s − 10·31-s − 35-s + 8·37-s + 2·41-s + 4·43-s − 4·47-s + 49-s − 4·53-s + 6·55-s − 6·59-s − 2·61-s − 4·65-s − 8·67-s − 8·71-s + 4·73-s − 6·77-s − 14·79-s − 4·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 1.80·11-s − 1.10·13-s − 0.485·17-s + 1.83·19-s − 0.208·23-s + 1/5·25-s − 0.371·29-s − 1.79·31-s − 0.169·35-s + 1.31·37-s + 0.312·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s − 0.549·53-s + 0.809·55-s − 0.781·59-s − 0.256·61-s − 0.496·65-s − 0.977·67-s − 0.949·71-s + 0.468·73-s − 0.683·77-s − 1.57·79-s − 0.439·83-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: \(1\)
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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.96418137853342, −13.36502471354710, −12.83156296148197, −12.41940738960627, −11.87115671821083, −11.46726401971580, −11.08407376551253, −10.34304294597077, −9.702194328342972, −9.444358735367005, −9.180640802692256, −8.612508304719428, −7.659235174789585, −7.378912323782479, −6.938052516163266, −6.258009257414043, −5.844879378644881, −5.340086336954185, −4.547984081151237, −4.213592816523466, −3.378085576360518, −3.047792232821096, −2.179329702314815, −1.587045345864003, −0.9641727531897025, 0, 0.9641727531897025, 1.587045345864003, 2.179329702314815, 3.047792232821096, 3.378085576360518, 4.213592816523466, 4.547984081151237, 5.340086336954185, 5.844879378644881, 6.258009257414043, 6.938052516163266, 7.378912323782479, 7.659235174789585, 8.612508304719428, 9.180640802692256, 9.444358735367005, 9.702194328342972, 10.34304294597077, 11.08407376551253, 11.46726401971580, 11.87115671821083, 12.41940738960627, 12.83156296148197, 13.36502471354710, 13.96418137853342

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