Properties

Label 2-388080-1.1-c1-0-127
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 − 3·13-s + 2·17-s − 8·19-s − 6·23-s + 25-s + 6·29-s + 9·31-s + 8·37-s − 8·41-s + 7·43-s + 4·47-s + 2·53-s + 55-s − 5·59-s + 14·61-s − 3·65-s + 4·67-s + 9·71-s − 73-s + 12·79-s + 9·83-s + 2·85-s + 7·89-s − 8·95-s + 6·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.301·11-s − 0.832·13-s + 0.485·17-s − 1.83·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.61·31-s + 1.31·37-s − 1.24·41-s + 1.06·43-s + 0.583·47-s + 0.274·53-s + 0.134·55-s − 0.650·59-s + 1.79·61-s − 0.372·65-s + 0.488·67-s + 1.06·71-s − 0.117·73-s + 1.35·79-s + 0.987·83-s + 0.216·85-s + 0.741·89-s − 0.820·95-s + 0.609·97-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

\(L(1)\) \(\approx\) \(3.122859410\)
\(L(\frac12)\) \(\approx\) \(3.122859410\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
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

−12.55461614291980, −11.99545526759440, −11.71272030863397, −11.07161712809023, −10.49859617032288, −10.22380097557365, −9.808150112344661, −9.414413008263072, −8.766017758232442, −8.342944056533249, −7.986998878279039, −7.500689875874604, −6.761835964071409, −6.378322467510756, −6.192611432472360, −5.473748271928418, −4.945641335838138, −4.433048619883254, −4.080755190203495, −3.445097786067534, −2.672932527649429, −2.297388023779589, −1.911702333342016, −0.9541088863373516, −0.5214365634282056, 0.5214365634282056, 0.9541088863373516, 1.911702333342016, 2.297388023779589, 2.672932527649429, 3.445097786067534, 4.080755190203495, 4.433048619883254, 4.945641335838138, 5.473748271928418, 6.192611432472360, 6.378322467510756, 6.761835964071409, 7.500689875874604, 7.986998878279039, 8.342944056533249, 8.766017758232442, 9.414413008263072, 9.808150112344661, 10.22380097557365, 10.49859617032288, 11.07161712809023, 11.71272030863397, 11.99545526759440, 12.55461614291980

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