Properties

Label 2-388080-1.1-c1-0-131
Degree $2$
Conductor $388080$
Sign $1$
Analytic cond. $3098.83$
Root an. cond. $55.6671$
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 − 11-s + 7·13-s − 2·17-s + 2·23-s + 25-s − 2·29-s + 3·31-s + 12·37-s − 6·41-s − 43-s − 10·47-s − 55-s + 7·59-s + 7·65-s + 4·67-s + 9·71-s + 9·73-s + 6·79-s + 11·83-s − 2·85-s − 7·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.301·11-s + 1.94·13-s − 0.485·17-s + 0.417·23-s + 1/5·25-s − 0.371·29-s + 0.538·31-s + 1.97·37-s − 0.937·41-s − 0.152·43-s − 1.45·47-s − 0.134·55-s + 0.911·59-s + 0.868·65-s + 0.488·67-s + 1.06·71-s + 1.05·73-s + 0.675·79-s + 1.20·83-s − 0.216·85-s − 0.741·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(388080\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(3098.83\)
Root analytic conductor: \(55.6671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 388080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.685696820\)
\(L(\frac12)\) \(\approx\) \(3.685696820\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + T \)
good13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−12.63446175364391, −11.94578635073520, −11.37240227792103, −11.10235254190546, −10.83091282716920, −10.19065243186169, −9.662086555920499, −9.420879530821638, −8.737096759490587, −8.365525115993609, −8.065906249601581, −7.478329399874357, −6.685799135541005, −6.476294460680163, −6.125547579580267, −5.383103910264545, −5.153625369655832, −4.413556297219580, −3.921296818091021, −3.449768917490382, −2.869691654994540, −2.294459663190067, −1.671329864641733, −1.106966812006244, −0.5325756948519706, 0.5325756948519706, 1.106966812006244, 1.671329864641733, 2.294459663190067, 2.869691654994540, 3.449768917490382, 3.921296818091021, 4.413556297219580, 5.153625369655832, 5.383103910264545, 6.125547579580267, 6.476294460680163, 6.685799135541005, 7.478329399874357, 8.065906249601581, 8.365525115993609, 8.737096759490587, 9.420879530821638, 9.662086555920499, 10.19065243186169, 10.83091282716920, 11.10235254190546, 11.37240227792103, 11.94578635073520, 12.63446175364391

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