Properties

Label 2-58800-1.1-c1-0-8
Degree $2$
Conductor $58800$
Sign $1$
Analytic cond. $469.520$
Root an. cond. $21.6684$
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  − 3-s + 9-s − 2·11-s − 6·13-s − 2·17-s + 4·23-s − 27-s − 8·31-s + 2·33-s + 2·37-s + 6·39-s − 2·41-s + 4·43-s − 8·47-s + 2·51-s + 6·53-s + 10·59-s − 2·61-s + 8·67-s − 4·69-s − 12·71-s + 4·73-s + 81-s − 4·83-s + 10·89-s + 8·93-s + 8·97-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.66·13-s − 0.485·17-s + 0.834·23-s − 0.192·27-s − 1.43·31-s + 0.348·33-s + 0.328·37-s + 0.960·39-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 0.280·51-s + 0.824·53-s + 1.30·59-s − 0.256·61-s + 0.977·67-s − 0.481·69-s − 1.42·71-s + 0.468·73-s + 1/9·81-s − 0.439·83-s + 1.05·89-s + 0.829·93-s + 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58800\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(469.520\)
Root analytic conductor: \(21.6684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{58800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5981187925\)
\(L(\frac12)\) \(\approx\) \(0.5981187925\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−14.46502048632916, −13.81281918075734, −13.06699055851846, −12.86485525121600, −12.40039445815615, −11.66929293318192, −11.39934076123971, −10.77037305243664, −10.21603178896424, −9.878003465886219, −9.165380144065562, −8.803255984035802, −7.914598343103223, −7.493416800314372, −7.021514662204049, −6.508159161516069, −5.755328623708870, −5.069033619309022, −4.995573523543875, −4.151551787589752, −3.481726484953341, −2.591950702057891, −2.227903324432214, −1.274470118903676, −0.2797460834183156, 0.2797460834183156, 1.274470118903676, 2.227903324432214, 2.591950702057891, 3.481726484953341, 4.151551787589752, 4.995573523543875, 5.069033619309022, 5.755328623708870, 6.508159161516069, 7.021514662204049, 7.493416800314372, 7.914598343103223, 8.803255984035802, 9.165380144065562, 9.878003465886219, 10.21603178896424, 10.77037305243664, 11.39934076123971, 11.66929293318192, 12.40039445815615, 12.86485525121600, 13.06699055851846, 13.81281918075734, 14.46502048632916

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