Properties

Label 2-338130-1.1-c1-0-3
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 $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 2·11-s − 13-s + 2·14-s + 16-s − 7·19-s − 20-s − 2·22-s + 2·23-s + 25-s + 26-s − 2·28-s + 2·29-s − 10·31-s − 32-s + 2·35-s + 3·37-s + 7·38-s + 40-s + 5·41-s − 6·43-s + 2·44-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 + 0.603·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.60·19-s − 0.223·20-s − 0.426·22-s + 0.417·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s + 0.371·29-s − 1.79·31-s − 0.176·32-s + 0.338·35-s + 0.493·37-s + 1.13·38-s + 0.158·40-s + 0.780·41-s − 0.914·43-s + 0.301·44-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: \(0\)
Selberg data: \((2,\ 338130,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1965959109\)
\(L(\frac12)\) \(\approx\) \(0.1965959109\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
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

−12.45213672367882, −12.23393008110098, −11.55097226747068, −11.16197826398263, −10.66439164547362, −10.43036957804281, −9.746183513402299, −9.280660374679356, −9.005632096935162, −8.598122562841965, −7.888424912465032, −7.632010454672567, −7.013609549822022, −6.591163746228303, −6.219909683482048, −5.741550416092103, −5.000521156263931, −4.410473677326160, −3.991883514892619, −3.328398492008749, −2.960902581065359, −2.211477028273097, −1.720261903319122, −1.005931204882889, −0.1439682889361424, 0.1439682889361424, 1.005931204882889, 1.720261903319122, 2.211477028273097, 2.960902581065359, 3.328398492008749, 3.991883514892619, 4.410473677326160, 5.000521156263931, 5.741550416092103, 6.219909683482048, 6.591163746228303, 7.013609549822022, 7.632010454672567, 7.888424912465032, 8.598122562841965, 9.005632096935162, 9.280660374679356, 9.746183513402299, 10.43036957804281, 10.66439164547362, 11.16197826398263, 11.55097226747068, 12.23393008110098, 12.45213672367882

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