Properties

Label 2-338130-1.1-c1-0-29
Degree $2$
Conductor $338130$
Sign $-1$
Analytic cond. $2699.98$
Root an. cond. $51.9613$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s − 3·11-s − 13-s + 2·14-s + 16-s + 5·19-s − 20-s + 3·22-s + 5·23-s + 25-s + 26-s − 2·28-s − 9·29-s − 31-s − 32-s + 2·35-s − 7·37-s − 5·38-s + 40-s − 6·41-s − 11·43-s − 3·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s − 0.904·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s + 1.14·19-s − 0.223·20-s + 0.639·22-s + 1.04·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 1.67·29-s − 0.179·31-s − 0.176·32-s + 0.338·35-s − 1.15·37-s − 0.811·38-s + 0.158·40-s − 0.937·41-s − 1.67·43-s − 0.452·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338130\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2699.98\)
Root analytic conductor: \(51.9613\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{338130} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 338130,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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

−12.91733380437017, −12.28851659985746, −11.71142508271040, −11.50468427723320, −10.96775989985626, −10.38463323135886, −10.11972091303611, −9.600715734454886, −9.197822106788349, −8.707561020952558, −8.267308559752819, −7.664674820272189, −7.273880140253106, −7.043267243154974, −6.400451517285280, −5.826260262029856, −5.236568701174129, −4.988895974195362, −4.218293314875372, −3.412365042742350, −3.143181864547954, −2.821726054471877, −1.754270467393261, −1.553777603387116, −0.4668042514198633, 0, 0.4668042514198633, 1.553777603387116, 1.754270467393261, 2.821726054471877, 3.143181864547954, 3.412365042742350, 4.218293314875372, 4.988895974195362, 5.236568701174129, 5.826260262029856, 6.400451517285280, 7.043267243154974, 7.273880140253106, 7.664674820272189, 8.267308559752819, 8.707561020952558, 9.197822106788349, 9.600715734454886, 10.11972091303611, 10.38463323135886, 10.96775989985626, 11.50468427723320, 11.71142508271040, 12.28851659985746, 12.91733380437017

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