Properties

Label 2-7488-1.1-c1-0-78
Degree $2$
Conductor $7488$
Sign $-1$
Analytic cond. $59.7919$
Root an. cond. $7.73252$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·7-s − 6·11-s + 13-s + 2·17-s + 6·19-s − 25-s − 6·29-s + 6·31-s − 4·35-s − 2·37-s + 10·41-s − 8·43-s − 6·47-s − 3·49-s + 6·53-s + 12·55-s − 6·59-s + 10·61-s − 2·65-s − 2·67-s + 14·71-s − 14·73-s − 12·77-s + 4·79-s + 6·83-s − 4·85-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s − 1.80·11-s + 0.277·13-s + 0.485·17-s + 1.37·19-s − 1/5·25-s − 1.11·29-s + 1.07·31-s − 0.676·35-s − 0.328·37-s + 1.56·41-s − 1.21·43-s − 0.875·47-s − 3/7·49-s + 0.824·53-s + 1.61·55-s − 0.781·59-s + 1.28·61-s − 0.248·65-s − 0.244·67-s + 1.66·71-s − 1.63·73-s − 1.36·77-s + 0.450·79-s + 0.658·83-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7488\)    =    \(2^{6} \cdot 3^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(59.7919\)
Root analytic conductor: \(7.73252\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7488,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.72311637943970507489745672120, −7.15351246089745902827498765258, −6.01751931907727312393576018481, −5.25874893030557723454431240338, −4.87531570308063726295785778280, −3.88536518722750035916842998102, −3.17455449304663684213943314812, −2.34100159140218790887026848642, −1.18343602639981650227699912137, 0, 1.18343602639981650227699912137, 2.34100159140218790887026848642, 3.17455449304663684213943314812, 3.88536518722750035916842998102, 4.87531570308063726295785778280, 5.25874893030557723454431240338, 6.01751931907727312393576018481, 7.15351246089745902827498765258, 7.72311637943970507489745672120

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