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Properties

Label 2-115920-1.1-c1-0-14
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 + 2·13-s + 2·17-s + 4·19-s − 23-s + 25-s − 6·29-s + 35-s − 10·37-s + 2·41-s − 4·43-s + 8·47-s + 49-s − 6·53-s − 4·55-s + 10·61-s + 2·65-s − 4·67-s + 14·73-s − 4·77-s + 4·79-s − 8·83-s + 2·85-s − 6·89-s + 2·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.11·29-s + 0.169·35-s − 1.64·37-s + 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.539·55-s + 1.28·61-s + 0.248·65-s − 0.488·67-s + 1.63·73-s − 0.455·77-s + 0.450·79-s − 0.878·83-s + 0.216·85-s − 0.635·89-s + 0.209·91-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

\(L(1)\) \(\approx\) \(2.423890162\)
\(L(\frac12)\) \(\approx\) \(2.423890162\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
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

−13.66250444304289, −13.14881683174794, −12.66381956739360, −12.24056860055419, −11.57951470774741, −11.18178874588093, −10.61146209908438, −10.27653805232361, −9.696092386774219, −9.267733903864679, −8.583506153656149, −8.220571602389105, −7.602558241786492, −7.240950210211136, −6.633416518767843, −5.811590989435595, −5.562889340058213, −5.085706268468280, −4.513095726754787, −3.599745047819077, −3.342056710431098, −2.493528344628342, −1.963460271964846, −1.299099558702424, −0.4839577847144021, 0.4839577847144021, 1.299099558702424, 1.963460271964846, 2.493528344628342, 3.342056710431098, 3.599745047819077, 4.513095726754787, 5.085706268468280, 5.562889340058213, 5.811590989435595, 6.633416518767843, 7.240950210211136, 7.602558241786492, 8.220571602389105, 8.583506153656149, 9.267733903864679, 9.696092386774219, 10.27653805232361, 10.61146209908438, 11.18178874588093, 11.57951470774741, 12.24056860055419, 12.66381956739360, 13.14881683174794, 13.66250444304289

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