Properties

Label 2-30960-1.1-c1-0-12
Degree $2$
Conductor $30960$
Sign $1$
Analytic cond. $247.216$
Root an. cond. $15.7231$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 4·11-s − 5·13-s + 8·17-s + 5·19-s + 8·23-s + 25-s + 3·29-s + 31-s − 35-s + 8·37-s + 11·41-s + 43-s + 10·47-s − 6·49-s + 2·53-s + 4·55-s + 3·61-s + 5·65-s − 11·67-s − 6·71-s − 11·73-s − 4·77-s + 7·79-s − 8·85-s + 16·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 1.20·11-s − 1.38·13-s + 1.94·17-s + 1.14·19-s + 1.66·23-s + 1/5·25-s + 0.557·29-s + 0.179·31-s − 0.169·35-s + 1.31·37-s + 1.71·41-s + 0.152·43-s + 1.45·47-s − 6/7·49-s + 0.274·53-s + 0.539·55-s + 0.384·61-s + 0.620·65-s − 1.34·67-s − 0.712·71-s − 1.28·73-s − 0.455·77-s + 0.787·79-s − 0.867·85-s + 1.69·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30960\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 43\)
Sign: $1$
Analytic conductor: \(247.216\)
Root analytic conductor: \(15.7231\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{30960} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 30960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.319762538\)
\(L(\frac12)\) \(\approx\) \(2.319762538\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
43 \( 1 - T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−15.05794580672635, −14.50595760354455, −14.29974984180021, −13.44785563614332, −12.93364718284548, −12.37905437126239, −11.98507052048273, −11.43859789930672, −10.76743831239264, −10.30848501831650, −9.704613232211912, −9.284051394320486, −8.478902460388285, −7.740866094999591, −7.488766684236265, −7.236850851325157, −6.042501168538947, −5.526544102645372, −4.923300003263480, −4.588114826486185, −3.557837818888938, −2.790334797897311, −2.609872453639040, −1.234565669969843, −0.6569361068696943, 0.6569361068696943, 1.234565669969843, 2.609872453639040, 2.790334797897311, 3.557837818888938, 4.588114826486185, 4.923300003263480, 5.526544102645372, 6.042501168538947, 7.236850851325157, 7.488766684236265, 7.740866094999591, 8.478902460388285, 9.284051394320486, 9.704613232211912, 10.30848501831650, 10.76743831239264, 11.43859789930672, 11.98507052048273, 12.37905437126239, 12.93364718284548, 13.44785563614332, 14.29974984180021, 14.50595760354455, 15.05794580672635

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