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Properties

Label 2-115920-1.1-c1-0-16
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 − 4·11-s + 4·13-s − 2·17-s − 4·19-s + 23-s + 25-s + 8·31-s + 35-s + 10·37-s + 2·41-s + 8·47-s + 49-s − 12·53-s + 4·55-s + 14·59-s + 8·61-s − 4·65-s + 4·67-s + 8·71-s + 12·73-s + 4·77-s − 2·79-s − 6·83-s + 2·85-s − 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.20·11-s + 1.10·13-s − 0.485·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.43·31-s + 0.169·35-s + 1.64·37-s + 0.312·41-s + 1.16·47-s + 1/7·49-s − 1.64·53-s + 0.539·55-s + 1.82·59-s + 1.02·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 1.40·73-s + 0.455·77-s − 0.225·79-s − 0.658·83-s + 0.216·85-s − 0.635·89-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

\(L(1)\) \(\approx\) \(1.919533596\)
\(L(\frac12)\) \(\approx\) \(1.919533596\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.42246193827840, −13.17768493612348, −12.64768460328575, −12.33436887224550, −11.50295601534342, −11.08224408426909, −10.86825221993698, −10.20681280687154, −9.725014211255703, −9.222920938283697, −8.397500822758926, −8.278792763812772, −7.852252623629098, −6.922017023131130, −6.754039316279127, −5.969091481729220, −5.666908851959174, −4.854748997317639, −4.346051207882188, −3.887065446444184, −3.172576105705398, −2.578691479816736, −2.131710373492397, −1.028232232874559, −0.4953767482664761, 0.4953767482664761, 1.028232232874559, 2.131710373492397, 2.578691479816736, 3.172576105705398, 3.887065446444184, 4.346051207882188, 4.854748997317639, 5.666908851959174, 5.969091481729220, 6.754039316279127, 6.922017023131130, 7.852252623629098, 8.278792763812772, 8.397500822758926, 9.222920938283697, 9.725014211255703, 10.20681280687154, 10.86825221993698, 11.08224408426909, 11.50295601534342, 12.33436887224550, 12.64768460328575, 13.17768493612348, 13.42246193827840

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