Properties

Label 2-936-1.1-c1-0-11
Degree $2$
Conductor $936$
Sign $-1$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·7-s − 4·11-s − 13-s − 2·19-s − 4·23-s − 25-s + 2·31-s − 4·35-s − 10·37-s − 2·41-s + 8·43-s − 3·49-s − 12·53-s + 8·55-s − 12·59-s − 6·61-s + 2·65-s − 6·67-s + 8·71-s − 2·73-s − 8·77-s + 12·79-s + 4·83-s − 14·89-s − 2·91-s + 4·95-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s − 1.20·11-s − 0.277·13-s − 0.458·19-s − 0.834·23-s − 1/5·25-s + 0.359·31-s − 0.676·35-s − 1.64·37-s − 0.312·41-s + 1.21·43-s − 3/7·49-s − 1.64·53-s + 1.07·55-s − 1.56·59-s − 0.768·61-s + 0.248·65-s − 0.733·67-s + 0.949·71-s − 0.234·73-s − 0.911·77-s + 1.35·79-s + 0.439·83-s − 1.48·89-s − 0.209·91-s + 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.709908242354085867198585893168, −8.560857309044347612087495782838, −7.917819582253129208914516956622, −7.40568873395535407748018257450, −6.16078101072896661105492972029, −5.08169261977535984530512865091, −4.35999893307754658375735542809, −3.20854141442725326453087823255, −1.93418903533460519772261561653, 0, 1.93418903533460519772261561653, 3.20854141442725326453087823255, 4.35999893307754658375735542809, 5.08169261977535984530512865091, 6.16078101072896661105492972029, 7.40568873395535407748018257450, 7.917819582253129208914516956622, 8.560857309044347612087495782838, 9.709908242354085867198585893168

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