↑

Properties

Label 2-30446-1.1-c1-0-7
Degree $2$
Conductor $30446$
Sign $-1$
Analytic cond. $243.112$
Root an. cond. $15.5920$
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-s − 3-s + 4-s + 3·5-s − 6-s + 7-s + 8-s − 2·9-s + 3·10-s − 3·11-s − 12-s − 13-s + 14-s − 3·15-s + 16-s − 6·17-s − 2·18-s − 4·19-s + 3·20-s − 21-s − 3·22-s + 4·23-s − 24-s + 4·25-s − 26-s + 5·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s − 0.904·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 1.45·17-s − 0.471·18-s − 0.917·19-s + 0.670·20-s − 0.218·21-s − 0.639·22-s + 0.834·23-s − 0.204·24-s + 4/5·25-s − 0.196·26-s + 0.962·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30446\)    =    \(2 \cdot 13 \cdot 1171\)
Sign: $-1$
Analytic conductor: \(243.112\)
Root analytic conductor: \(15.5920\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 30446,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
13 \( 1 + T \)
1171 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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

−15.18778539076645, −14.95166400394216, −14.09320969022272, −13.80482031583875, −13.32474549014431, −12.79721625174611, −12.38355707171167, −11.51257184600566, −11.21436870876167, −10.62218725290017, −10.23426694066263, −9.550909663508386, −8.866563625555077, −8.308795354686601, −7.708903092619232, −6.715517129183480, −6.308601128059462, −6.059150784250054, −5.179968589275410, −4.800082936803414, −4.430743962004849, −3.104146526146259, −2.577106858101356, −2.131606190480443, −1.165285613582469, 0, 1.165285613582469, 2.131606190480443, 2.577106858101356, 3.104146526146259, 4.430743962004849, 4.800082936803414, 5.179968589275410, 6.059150784250054, 6.308601128059462, 6.715517129183480, 7.708903092619232, 8.308795354686601, 8.866563625555077, 9.550909663508386, 10.23426694066263, 10.62218725290017, 11.21436870876167, 11.51257184600566, 12.38355707171167, 12.79721625174611, 13.32474549014431, 13.80482031583875, 14.09320969022272, 14.95166400394216, 15.18778539076645

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