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Properties

Label 2-115920-1.1-c1-0-28
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 − 2·13-s − 6·17-s + 4·19-s + 23-s + 25-s + 10·29-s + 4·31-s − 35-s + 10·37-s + 2·41-s + 4·43-s + 49-s + 6·53-s − 4·59-s − 6·61-s − 2·65-s − 4·67-s + 8·71-s + 10·73-s − 16·79-s + 16·83-s − 6·85-s + 10·89-s + 2·91-s + 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.85·29-s + 0.718·31-s − 0.169·35-s + 1.64·37-s + 0.312·41-s + 0.609·43-s + 1/7·49-s + 0.824·53-s − 0.520·59-s − 0.768·61-s − 0.248·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s − 1.80·79-s + 1.75·83-s − 0.650·85-s + 1.05·89-s + 0.209·91-s + 0.410·95-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

\(L(1)\) \(\approx\) \(2.716660951\)
\(L(\frac12)\) \(\approx\) \(2.716660951\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.61740563486731, −13.23774052825756, −12.60002427957238, −12.20526199527480, −11.70915642371699, −11.14326240249559, −10.67711212031483, −10.12886810490704, −9.700962700383287, −9.193095071429384, −8.838323189520874, −8.128477480267533, −7.669115089096604, −7.059632285366903, −6.433325146909684, −6.286443550029902, −5.497064666917473, −4.892582749482512, −4.469874186805842, −3.910818698592028, −2.922992449697568, −2.705274334569335, −2.071697306992922, −1.123102568583102, −0.5584006733649888, 0.5584006733649888, 1.123102568583102, 2.071697306992922, 2.705274334569335, 2.922992449697568, 3.910818698592028, 4.469874186805842, 4.892582749482512, 5.497064666917473, 6.286443550029902, 6.433325146909684, 7.059632285366903, 7.669115089096604, 8.128477480267533, 8.838323189520874, 9.193095071429384, 9.700962700383287, 10.12886810490704, 10.67711212031483, 11.14326240249559, 11.70915642371699, 12.20526199527480, 12.60002427957238, 13.23774052825756, 13.61740563486731

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