Properties

Label 2-9338-1.1-c1-0-213
Degree $2$
Conductor $9338$
Sign $-1$
Analytic cond. $74.5643$
Root an. cond. $8.63506$
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 − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 9-s + 4·11-s − 2·12-s − 2·13-s − 14-s + 16-s + 4·17-s + 18-s + 2·21-s + 4·22-s + 23-s − 2·24-s − 5·25-s − 2·26-s + 4·27-s − 28-s − 29-s − 10·31-s + 32-s − 8·33-s + 4·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.577·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.436·21-s + 0.852·22-s + 0.208·23-s − 0.408·24-s − 25-s − 0.392·26-s + 0.769·27-s − 0.188·28-s − 0.185·29-s − 1.79·31-s + 0.176·32-s − 1.39·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9338\)    =    \(2 \cdot 7 \cdot 23 \cdot 29\)
Sign: $-1$
Analytic conductor: \(74.5643\)
Root analytic conductor: \(8.63506\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{9338} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9338,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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

−7.04702318268006394021919158000, −6.58145978867175709899295156507, −5.84542791087276500355197405296, −5.41899955045890602081420861549, −4.75632996696133313983499645152, −3.80336946502404333745381531113, −3.37129921117625599375688138710, −2.15383403670028212753028308774, −1.20919625717538211058601405515, 0, 1.20919625717538211058601405515, 2.15383403670028212753028308774, 3.37129921117625599375688138710, 3.80336946502404333745381531113, 4.75632996696133313983499645152, 5.41899955045890602081420861549, 5.84542791087276500355197405296, 6.58145978867175709899295156507, 7.04702318268006394021919158000

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