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Properties

Label 2-2736-228.227-c0-0-8
Degree $2$
Conductor $2736$
Sign $0.0917 + 0.995i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.517i·5-s − i·7-s + 0.517·11-s − 1.93i·17-s + i·19-s − 1.41·23-s + 0.732·25-s − 0.517·35-s − 1.73i·43-s − 1.93·47-s − 0.267i·55-s + 1.73·61-s − 73-s − 0.517i·77-s + 1.41·83-s + ⋯
L(s)  = 1  − 0.517i·5-s − i·7-s + 0.517·11-s − 1.93i·17-s + i·19-s − 1.41·23-s + 0.732·25-s − 0.517·35-s − 1.73i·43-s − 1.93·47-s − 0.267i·55-s + 1.73·61-s − 73-s − 0.517i·77-s + 1.41·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.0917 + 0.995i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ 0.0917 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.142358004\)
\(L(\frac12)\) \(\approx\) \(1.142358004\)
\(L(1)\) 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 \( 1 - iT \)
good5 \( 1 + 0.517iT - T^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 - 0.517T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.93iT - T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 + 1.93T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.73T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
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

−8.802613125151395881222116633159, −8.077710667137686646634459608322, −7.27117413827582439974439281521, −6.71218471243661975243681602585, −5.67482753727372877335976260354, −4.85856662747130656477379688143, −4.08997092199353151072725277805, −3.29171146360642447334460619284, −1.96585402786050716035646503160, −0.75260675886657805150076381824, 1.63991485043357070971683986163, 2.57390535570138044752414877232, 3.53268164377710775151331755105, 4.42295179270097513805726384324, 5.42986620025226661133113198453, 6.30821089635726170146844717923, 6.61331032585000722876570364095, 7.84635489511807641632945970701, 8.420941329034339995430898391752, 9.117420526265624875008469758275

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