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Properties

Label 2-115920-1.1-c1-0-10
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 + 3·11-s − 4·13-s + 5·19-s + 23-s + 25-s − 4·29-s + 8·31-s + 35-s + 4·37-s − 9·41-s + 6·43-s + 47-s + 49-s − 11·53-s − 3·55-s − 9·59-s + 61-s + 4·65-s + 10·67-s + 12·71-s − 3·77-s − 4·79-s − 6·83-s + 12·89-s + 4·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 0.904·11-s − 1.10·13-s + 1.14·19-s + 0.208·23-s + 1/5·25-s − 0.742·29-s + 1.43·31-s + 0.169·35-s + 0.657·37-s − 1.40·41-s + 0.914·43-s + 0.145·47-s + 1/7·49-s − 1.51·53-s − 0.404·55-s − 1.17·59-s + 0.128·61-s + 0.496·65-s + 1.22·67-s + 1.42·71-s − 0.341·77-s − 0.450·79-s − 0.658·83-s + 1.27·89-s + 0.419·91-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.783514556\)
\(L(\frac12)\) \(\approx\) \(1.783514556\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−13.68222509624670, −13.09950184923009, −12.51046284029081, −12.04049184583813, −11.87847020099357, −11.16196830608973, −10.83545915483146, −10.01454512396683, −9.576241689125927, −9.422665349747519, −8.708739830066121, −8.036995130718925, −7.694537218163221, −7.120302646726436, −6.580312218683518, −6.253004636675613, −5.276550303456075, −5.117996789334333, −4.236709767118534, −3.939900884007847, −3.048173958019575, −2.839623037185281, −1.892755070672096, −1.177502094807703, −0.4430579839990738, 0.4430579839990738, 1.177502094807703, 1.892755070672096, 2.839623037185281, 3.048173958019575, 3.939900884007847, 4.236709767118534, 5.117996789334333, 5.276550303456075, 6.253004636675613, 6.580312218683518, 7.120302646726436, 7.694537218163221, 8.036995130718925, 8.708739830066121, 9.422665349747519, 9.576241689125927, 10.01454512396683, 10.83545915483146, 11.16196830608973, 11.87847020099357, 12.04049184583813, 12.51046284029081, 13.09950184923009, 13.68222509624670

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