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Properties

Label 2-388080-1.1-c1-0-117
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 + 6·13-s − 7·17-s − 5·19-s − 23-s + 25-s + 5·29-s − 8·31-s − 2·37-s + 12·41-s + 11·43-s − 8·47-s + 11·53-s + 55-s + 5·59-s − 7·61-s + 6·65-s + 2·67-s + 12·71-s − 4·73-s + 10·79-s + 83-s − 7·85-s + 15·89-s − 5·95-s − 3·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.301·11-s + 1.66·13-s − 1.69·17-s − 1.14·19-s − 0.208·23-s + 1/5·25-s + 0.928·29-s − 1.43·31-s − 0.328·37-s + 1.87·41-s + 1.67·43-s − 1.16·47-s + 1.51·53-s + 0.134·55-s + 0.650·59-s − 0.896·61-s + 0.744·65-s + 0.244·67-s + 1.42·71-s − 0.468·73-s + 1.12·79-s + 0.109·83-s − 0.759·85-s + 1.58·89-s − 0.512·95-s − 0.304·97-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

\(L(1)\) \(\approx\) \(3.177662635\)
\(L(\frac12)\) \(\approx\) \(3.177662635\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−12.58421311358750, −12.03795294416349, −11.42013327554845, −10.95040605144621, −10.72815063411365, −10.49803414368102, −9.539955250138977, −9.259503649673907, −8.885894571091616, −8.413207960651372, −8.091545069709227, −7.316260286551118, −6.851825024727082, −6.340540264392946, −6.128597042749336, −5.603989016542416, −4.971054712600199, −4.285452946770601, −4.054318939087079, −3.552863364827040, −2.761248529441984, −2.184314264030186, −1.863501865424398, −1.050834750279532, −0.4954888011431058, 0.4954888011431058, 1.050834750279532, 1.863501865424398, 2.184314264030186, 2.761248529441984, 3.552863364827040, 4.054318939087079, 4.285452946770601, 4.971054712600199, 5.603989016542416, 6.128597042749336, 6.340540264392946, 6.851825024727082, 7.316260286551118, 8.091545069709227, 8.413207960651372, 8.885894571091616, 9.259503649673907, 9.539955250138977, 10.49803414368102, 10.72815063411365, 10.95040605144621, 11.42013327554845, 12.03795294416349, 12.58421311358750

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