↑

Properties

Label 2-5616-12.11-c1-0-44
Degree $2$
Conductor $5616$
Sign $0.866 + 0.5i$
Analytic cond. $44.8439$
Root an. cond. $6.69656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.792i·5-s − 2.52i·7-s − 6.37·11-s + 13-s + 3.46i·17-s − 2.37·23-s + 4.37·25-s − 4.10i·29-s + 9.15i·31-s + 2·35-s + 6.37·37-s − 10.0i·41-s + 9.94i·43-s − 9.37·47-s + 0.627·49-s + ⋯
L(s)  = 1  + 0.354i·5-s − 0.954i·7-s − 1.92·11-s + 0.277·13-s + 0.840i·17-s − 0.494·23-s + 0.874·25-s − 0.763i·29-s + 1.64i·31-s + 0.338·35-s + 1.04·37-s − 1.57i·41-s + 1.51i·43-s − 1.36·47-s + 0.0896·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5616\)    =    \(2^{4} \cdot 3^{3} \cdot 13\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(44.8439\)
Root analytic conductor: \(6.69656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5616} (2159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5616,\ (\ :1/2),\ 0.866 + 0.5i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.429546530\)
\(L(\frac12)\) \(\approx\) \(1.429546530\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - T \)
good5 \( 1 - 0.792iT - 5T^{2} \)
7 \( 1 + 2.52iT - 7T^{2} \)
11 \( 1 + 6.37T + 11T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 2.37T + 23T^{2} \)
29 \( 1 + 4.10iT - 29T^{2} \)
31 \( 1 - 9.15iT - 31T^{2} \)
37 \( 1 - 6.37T + 37T^{2} \)
41 \( 1 + 10.0iT - 41T^{2} \)
43 \( 1 - 9.94iT - 43T^{2} \)
47 \( 1 + 9.37T + 47T^{2} \)
53 \( 1 - 7.86iT - 53T^{2} \)
59 \( 1 - 0.255T + 59T^{2} \)
61 \( 1 + 1.37T + 61T^{2} \)
67 \( 1 + 11.6iT - 67T^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 - 17.7T + 83T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.912828407645525134422559358284, −7.54850241085772090654753479662, −6.65382438655585538583175545651, −6.01739370903721101554212976913, −5.10056301256549658597032475108, −4.48576731401589168916260935695, −3.50843889931881145051047510607, −2.83378617745267196617998074800, −1.82139909160886172055573472588, −0.53354969084366951278357341437, 0.71961544156739551216017756219, 2.23979148458127291726117834200, 2.65620600388772793565056424732, 3.67047499229173842107653734945, 4.90259462206184786413003099077, 5.16883064637737594688699588471, 5.92722678136653075814233224836, 6.74380305894638384305888203165, 7.71798486540410531835551180907, 8.144754132766878976889075186192

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