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Properties

Label 2-6336-1.1-c1-0-99
Degree $2$
Conductor $6336$
Sign $1$
Analytic cond. $50.5932$
Root an. cond. $7.11289$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s + 11-s − 6·13-s − 6·17-s − 8·19-s − 25-s − 6·29-s + 8·35-s − 6·37-s + 10·41-s − 8·43-s + 9·49-s + 6·53-s − 2·55-s − 4·59-s + 2·61-s + 12·65-s − 12·67-s − 8·71-s + 2·73-s − 4·77-s + 4·79-s + 12·83-s + 12·85-s + 6·89-s + 24·91-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s + 0.301·11-s − 1.66·13-s − 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.35·35-s − 0.986·37-s + 1.56·41-s − 1.21·43-s + 9/7·49-s + 0.824·53-s − 0.269·55-s − 0.520·59-s + 0.256·61-s + 1.48·65-s − 1.46·67-s − 0.949·71-s + 0.234·73-s − 0.455·77-s + 0.450·79-s + 1.31·83-s + 1.30·85-s + 0.635·89-s + 2.51·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6336\)    =    \(2^{6} \cdot 3^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(50.5932\)
Root analytic conductor: \(7.11289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 6336,\ (\ :1/2),\ 1)\)

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

−7.26761467471206446277452494394, −6.64789914404096307221037476012, −6.16512151881297020294415434382, −5.05923018649433147904755872370, −4.21904055084523704623394988253, −3.77935606629209915642007051847, −2.73981822679906449127385775603, −2.06536264866853690522154791248, 0, 0, 2.06536264866853690522154791248, 2.73981822679906449127385775603, 3.77935606629209915642007051847, 4.21904055084523704623394988253, 5.05923018649433147904755872370, 6.16512151881297020294415434382, 6.64789914404096307221037476012, 7.26761467471206446277452494394

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