Properties

Label 2-2736-57.56-c1-0-20
Degree $2$
Conductor $2736$
Sign $0.577 - 0.816i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.09i·5-s + 4.77·7-s + 2.91i·11-s − 6.02i·17-s + 4.35·19-s + 8.99i·23-s + 0.614·25-s + 10.0i·35-s + 10.8·43-s − 13.6i·47-s + 15.8·49-s − 6.10·55-s − 15.1·61-s − 16.8·73-s + 13.9i·77-s + ⋯
L(s)  = 1  + 0.936i·5-s + 1.80·7-s + 0.879i·11-s − 1.46i·17-s + 1.00·19-s + 1.87i·23-s + 0.122·25-s + 1.69i·35-s + 1.65·43-s − 1.99i·47-s + 2.26·49-s − 0.823·55-s − 1.94·61-s − 1.96·73-s + 1.58i·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.404937756\)
\(L(\frac12)\) \(\approx\) \(2.404937756\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.35T \)
good5 \( 1 - 2.09iT - 5T^{2} \)
7 \( 1 - 4.77T + 7T^{2} \)
11 \( 1 - 2.91iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 6.02iT - 17T^{2} \)
23 \( 1 - 8.99iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 13.6iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15.1T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16.8T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.024431210224107460036236832670, −7.922742132541417218949887833523, −7.35996012018145517498700575027, −7.05382168720677028610688038542, −5.63990486041635276370754726370, −5.10051312411496647433089888011, −4.29902890448271496276615308865, −3.18057909434203012996303894704, −2.23655524628980052958865653295, −1.26898684025014324042785367295, 0.920245764446407294774929266868, 1.67370350017004768666386418413, 2.92039920803672267573712557934, 4.32981354974515683460944309314, 4.60323126173634689588235301001, 5.60208358839565960170291498146, 6.17196468118969760651987564401, 7.52387153089481863999228803428, 8.042823841081682600836472361425, 8.700856533013482378185928879209

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