Properties

Label 2-2736-1.1-c1-0-7
Degree $2$
Conductor $2736$
Sign $1$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.91·5-s + 4.32·7-s − 0.0827·11-s − 7.02·13-s + 5.10·17-s + 19-s − 7.83·23-s + 10.3·25-s + 2.81·29-s + 6·31-s − 16.9·35-s + 4·37-s − 1.02·41-s − 0.324·43-s − 6.73·47-s + 11.6·49-s − 8.85·53-s + 0.324·55-s + 14.0·59-s + 9.34·61-s + 27.5·65-s − 1.62·67-s + 1.62·71-s + 3.30·73-s − 0.357·77-s + 13.0·79-s − 6.81·83-s + ⋯
L(s)  = 1  − 1.75·5-s + 1.63·7-s − 0.0249·11-s − 1.94·13-s + 1.23·17-s + 0.229·19-s − 1.63·23-s + 2.06·25-s + 0.522·29-s + 1.07·31-s − 2.86·35-s + 0.657·37-s − 0.159·41-s − 0.0494·43-s − 0.981·47-s + 1.67·49-s − 1.21·53-s + 0.0436·55-s + 1.82·59-s + 1.19·61-s + 3.41·65-s − 0.198·67-s + 0.193·71-s + 0.386·73-s − 0.0407·77-s + 1.46·79-s − 0.747·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.306839007\)
\(L(\frac12)\) \(\approx\) \(1.306839007\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 - T \)
good5 \( 1 + 3.91T + 5T^{2} \)
7 \( 1 - 4.32T + 7T^{2} \)
11 \( 1 + 0.0827T + 11T^{2} \)
13 \( 1 + 7.02T + 13T^{2} \)
17 \( 1 - 5.10T + 17T^{2} \)
23 \( 1 + 7.83T + 23T^{2} \)
29 \( 1 - 2.81T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 1.02T + 41T^{2} \)
43 \( 1 + 0.324T + 43T^{2} \)
47 \( 1 + 6.73T + 47T^{2} \)
53 \( 1 + 8.85T + 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 - 9.34T + 61T^{2} \)
67 \( 1 + 1.62T + 67T^{2} \)
71 \( 1 - 1.62T + 71T^{2} \)
73 \( 1 - 3.30T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + 6.81T + 83T^{2} \)
89 \( 1 - 1.18T + 89T^{2} \)
97 \( 1 + 5.66T + 97T^{2} \)
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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

−8.321844726316150612943765951997, −8.036955873675029182231850980071, −7.61943206764324692940821498562, −6.84365789378159225727909678767, −5.45592564043770390255880050309, −4.73173280935912438099276100532, −4.27174954043337402186507663683, −3.22189354944770394763854600378, −2.11307061209173312415203079123, −0.70946273429085315645708596144, 0.70946273429085315645708596144, 2.11307061209173312415203079123, 3.22189354944770394763854600378, 4.27174954043337402186507663683, 4.73173280935912438099276100532, 5.45592564043770390255880050309, 6.84365789378159225727909678767, 7.61943206764324692940821498562, 8.036955873675029182231850980071, 8.321844726316150612943765951997

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