Properties

Label 2-115920-1.1-c1-0-114
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 + 2·11-s + 2·13-s − 2·17-s + 4·19-s − 23-s + 25-s − 8·29-s − 2·31-s − 35-s + 4·37-s − 2·41-s + 8·43-s + 12·47-s + 49-s − 10·53-s + 2·55-s − 4·59-s − 8·61-s + 2·65-s + 8·67-s + 4·71-s − 6·73-s − 2·77-s − 8·79-s + 6·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.603·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.48·29-s − 0.359·31-s − 0.169·35-s + 0.657·37-s − 0.312·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s − 1.37·53-s + 0.269·55-s − 0.520·59-s − 1.02·61-s + 0.248·65-s + 0.977·67-s + 0.474·71-s − 0.702·73-s − 0.227·77-s − 0.900·79-s + 0.658·83-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: $\chi_{115920} (1, \cdot )$
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−13.92624267143550, −13.31263989643118, −12.98071567317507, −12.37643515840107, −12.00716125437868, −11.26201108851363, −11.01659910282378, −10.52339519303670, −9.738271340941728, −9.465107486696100, −9.045690057153847, −8.584558091651615, −7.752037905530322, −7.446223302942209, −6.823896158696620, −6.253902403038743, −5.811963415418465, −5.412914900774027, −4.611773602836727, −4.042467215430762, −3.561511355444279, −2.906455223839839, −2.255569919953157, −1.558094075541358, −0.9594432262854601, 0, 0.9594432262854601, 1.558094075541358, 2.255569919953157, 2.906455223839839, 3.561511355444279, 4.042467215430762, 4.611773602836727, 5.412914900774027, 5.811963415418465, 6.253902403038743, 6.823896158696620, 7.446223302942209, 7.752037905530322, 8.584558091651615, 9.045690057153847, 9.465107486696100, 9.738271340941728, 10.52339519303670, 11.01659910282378, 11.26201108851363, 12.00716125437868, 12.37643515840107, 12.98071567317507, 13.31263989643118, 13.92624267143550

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