Properties

Label 2-388080-1.1-c1-0-142
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s − 2·13-s − 3·17-s − 7·19-s − 3·23-s + 25-s + 3·29-s − 4·31-s − 10·37-s + 7·43-s − 9·53-s − 55-s − 3·59-s + 61-s + 2·65-s − 2·67-s − 12·71-s + 4·73-s − 2·79-s + 3·83-s + 3·85-s + 9·89-s + 7·95-s + 19·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 0.554·13-s − 0.727·17-s − 1.60·19-s − 0.625·23-s + 1/5·25-s + 0.557·29-s − 0.718·31-s − 1.64·37-s + 1.06·43-s − 1.23·53-s − 0.134·55-s − 0.390·59-s + 0.128·61-s + 0.248·65-s − 0.244·67-s − 1.42·71-s + 0.468·73-s − 0.225·79-s + 0.329·83-s + 0.325·85-s + 0.953·89-s + 0.718·95-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-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: \(1\)
Selberg data: \((2,\ 388080,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−12.61095207173790, −12.12668078717587, −11.98339446962800, −11.26041631948307, −10.88973899257152, −10.43904275197832, −10.16328292916062, −9.352439997938007, −9.094512944934460, −8.555658848922892, −8.252735210652422, −7.568301223931862, −7.254653335940992, −6.694199236209839, −6.229739151722453, −5.884506745875283, −5.087887677143731, −4.624906197356781, −4.317845389017622, −3.669609952466283, −3.268834076448109, −2.470496898927349, −2.046577715832921, −1.532222353973838, −0.5505035573704109, 0, 0.5505035573704109, 1.532222353973838, 2.046577715832921, 2.470496898927349, 3.268834076448109, 3.669609952466283, 4.317845389017622, 4.624906197356781, 5.087887677143731, 5.884506745875283, 6.229739151722453, 6.694199236209839, 7.254653335940992, 7.568301223931862, 8.252735210652422, 8.555658848922892, 9.094512944934460, 9.352439997938007, 10.16328292916062, 10.43904275197832, 10.88973899257152, 11.26041631948307, 11.98339446962800, 12.12668078717587, 12.61095207173790

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