Properties

Degree $2$
Conductor $3136$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s − 2·9-s + 3·11-s − 2·13-s + 3·15-s + 3·17-s + 19-s + 3·23-s + 4·25-s + 5·27-s + 6·29-s − 7·31-s − 3·33-s + 37-s + 2·39-s + 6·41-s + 4·43-s + 6·45-s − 9·47-s − 3·51-s − 3·53-s − 9·55-s − 57-s − 9·59-s + 61-s + 6·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 2/3·9-s + 0.904·11-s − 0.554·13-s + 0.774·15-s + 0.727·17-s + 0.229·19-s + 0.625·23-s + 4/5·25-s + 0.962·27-s + 1.11·29-s − 1.25·31-s − 0.522·33-s + 0.164·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.894·45-s − 1.31·47-s − 0.420·51-s − 0.412·53-s − 1.21·55-s − 0.132·57-s − 1.17·59-s + 0.128·61-s + 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{3136} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−8.227904921128916434268724018455, −7.56240616311991526194210715543, −6.87229450800291080599835044327, −6.05313952980566733119414581451, −5.18634468459653321605027216331, −4.42986893621370772138711109774, −3.57954413035331597786593448348, −2.80019703611351490822885633419, −1.16455596775948714060930265899, 0, 1.16455596775948714060930265899, 2.80019703611351490822885633419, 3.57954413035331597786593448348, 4.42986893621370772138711109774, 5.18634468459653321605027216331, 6.05313952980566733119414581451, 6.87229450800291080599835044327, 7.56240616311991526194210715543, 8.227904921128916434268724018455

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