↑

Properties

Label 2-2352-12.11-c1-0-51
Degree $2$
Conductor $2352$
Sign $0.866 + 0.5i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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  + 1.73i·3-s − 2.99·9-s − 5·13-s − 8.66i·19-s + 5·25-s − 5.19i·27-s − 1.73i·31-s + 11·37-s − 8.66i·39-s − 1.73i·43-s + 15·57-s + 14·61-s − 15.5i·67-s − 17·73-s + 8.66i·75-s + ⋯
L(s)  = 1  + 0.999i·3-s − 0.999·9-s − 1.38·13-s − 1.98i·19-s + 25-s − 0.999i·27-s − 0.311i·31-s + 1.80·37-s − 1.38i·39-s − 0.264i·43-s + 1.98·57-s + 1.79·61-s − 1.90i·67-s − 1.98·73-s + 0.999i·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.866 + 0.5i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.200781431\)
\(L(\frac12)\) \(\approx\) \(1.200781431\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 8.66iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 - 11T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 15.5iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 17T + 73T^{2} \)
79 \( 1 - 5.19iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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

−9.128691754364935088618471614736, −8.307683222779761480918317817608, −7.35967397948112898795965544474, −6.63118179842720350533181130416, −5.55787409394453839196742916217, −4.80281379300049063229030734441, −4.31676690728583145434139301925, −3.02176732685718534719438412126, −2.43892206883936286564975938291, −0.45385272928936839422177516881, 1.09350484800777772948870216775, 2.21038837821762574307790706485, 3.04069899470076543900225480315, 4.23017086447455815019264309753, 5.28929257871637349274120356145, 5.98178364255801151628703237848, 6.83226999443266191953270445749, 7.53605700383819288624992003235, 8.103250058971273347599610145631, 8.896267669228008254535555175692

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