Properties

Label 2-2736-19.18-c0-0-3
Degree $2$
Conductor $2736$
Sign $1$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·5-s − 7-s + 1.73·11-s − 1.73·17-s + 19-s + 1.99·25-s − 1.73·35-s + 43-s − 1.73·47-s + 2.99·55-s + 61-s − 73-s − 1.73·77-s − 2.99·85-s + 1.73·95-s + 1.73·119-s + ⋯
L(s)  = 1  + 1.73·5-s − 7-s + 1.73·11-s − 1.73·17-s + 19-s + 1.99·25-s − 1.73·35-s + 43-s − 1.73·47-s + 2.99·55-s + 61-s − 73-s − 1.73·77-s − 2.99·85-s + 1.73·95-s + 1.73·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (721, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.620659815\)
\(L(\frac12)\) \(\approx\) \(1.620659815\)
\(L(1)\) 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 \( 1 \)
19 \( 1 - T \)
good5 \( 1 - 1.73T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - 1.73T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.73T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + 1.73T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.235990735237979540840943919229, −8.619410630756537695021763134250, −7.15794941416174872333860017999, −6.46591072558838029086479624716, −6.23514015606536678206407764949, −5.26621118161322420950977046831, −4.27314550724798512049580917546, −3.25098285817593569644123243872, −2.27302423973733520947508430752, −1.33300774873847751313801934917, 1.33300774873847751313801934917, 2.27302423973733520947508430752, 3.25098285817593569644123243872, 4.27314550724798512049580917546, 5.26621118161322420950977046831, 6.23514015606536678206407764949, 6.46591072558838029086479624716, 7.15794941416174872333860017999, 8.619410630756537695021763134250, 9.235990735237979540840943919229

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