Properties

Label 2-9360-1.1-c1-0-81
Degree $2$
Conductor $9360$
Sign $-1$
Analytic cond. $74.7399$
Root an. cond. $8.64522$
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  − 5-s + 7-s − 3·11-s + 13-s + 3·17-s + 4·19-s − 9·23-s + 25-s + 6·29-s − 2·31-s − 35-s − 37-s + 3·41-s − 2·43-s − 6·47-s − 6·49-s − 9·53-s + 3·55-s − 12·59-s + 5·61-s − 65-s + 4·67-s + 9·71-s + 14·73-s − 3·77-s + 7·79-s − 3·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.904·11-s + 0.277·13-s + 0.727·17-s + 0.917·19-s − 1.87·23-s + 1/5·25-s + 1.11·29-s − 0.359·31-s − 0.169·35-s − 0.164·37-s + 0.468·41-s − 0.304·43-s − 0.875·47-s − 6/7·49-s − 1.23·53-s + 0.404·55-s − 1.56·59-s + 0.640·61-s − 0.124·65-s + 0.488·67-s + 1.06·71-s + 1.63·73-s − 0.341·77-s + 0.787·79-s − 0.325·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9360\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(74.7399\)
Root analytic conductor: \(8.64522\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9360,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
13 \( 1 - T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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

−7.60509129102512478740837795731, −6.67416512992499470665466656212, −5.98718265717636994106082741103, −5.21634311676118710669400472269, −4.68886628892012665231862128899, −3.74278738717854090919228621716, −3.13821906383307509599404373658, −2.18601208342146476185155948261, −1.20860568369192401347701347760, 0, 1.20860568369192401347701347760, 2.18601208342146476185155948261, 3.13821906383307509599404373658, 3.74278738717854090919228621716, 4.68886628892012665231862128899, 5.21634311676118710669400472269, 5.98718265717636994106082741103, 6.67416512992499470665466656212, 7.60509129102512478740837795731

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