Properties

Label 2-338130-1.1-c1-0-35
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 + 4·7-s − 8-s + 10-s − 2·11-s − 13-s − 4·14-s + 16-s + 6·19-s − 20-s + 2·22-s + 6·23-s + 25-s + 26-s + 4·28-s + 2·29-s + 6·31-s − 32-s − 4·35-s + 2·37-s − 6·38-s + 40-s + 10·41-s − 10·43-s − 2·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.603·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 1.37·19-s − 0.223·20-s + 0.426·22-s + 1.25·23-s + 1/5·25-s + 0.196·26-s + 0.755·28-s + 0.371·29-s + 1.07·31-s − 0.176·32-s − 0.676·35-s + 0.328·37-s − 0.973·38-s + 0.158·40-s + 1.56·41-s − 1.52·43-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

\(L(1)\) \(\approx\) \(2.475251217\)
\(L(\frac12)\) \(\approx\) \(2.475251217\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−12.31428553116302, −11.92492201268380, −11.74346613859510, −11.09827443621922, −10.69317951301863, −10.57746880982889, −9.737672931543167, −9.373343324633453, −8.910367353139103, −8.320309252509789, −7.999964249836999, −7.599139464486610, −7.268006969516623, −6.737340209108056, −6.034901596372335, −5.437880719036577, −5.080493057869114, −4.544079277785941, −4.168134665205943, −3.188931839520367, −2.876913637464587, −2.316379532068483, −1.509035180116162, −1.101583118726196, −0.5215180710797357, 0.5215180710797357, 1.101583118726196, 1.509035180116162, 2.316379532068483, 2.876913637464587, 3.188931839520367, 4.168134665205943, 4.544079277785941, 5.080493057869114, 5.437880719036577, 6.034901596372335, 6.737340209108056, 7.268006969516623, 7.599139464486610, 7.999964249836999, 8.320309252509789, 8.910367353139103, 9.373343324633453, 9.737672931543167, 10.57746880982889, 10.69317951301863, 11.09827443621922, 11.74346613859510, 11.92492201268380, 12.31428553116302

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