Properties

Label 2-588-196.187-c1-0-38
Degree $2$
Conductor $588$
Sign $0.874 + 0.485i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
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  + (−1.37 + 0.327i)2-s + (0.988 + 0.149i)3-s + (1.78 − 0.900i)4-s + (3.55 − 1.39i)5-s + (−1.40 + 0.118i)6-s + (−2.21 − 1.44i)7-s + (−2.16 + 1.82i)8-s + (0.955 + 0.294i)9-s + (−4.43 + 3.08i)10-s + (−0.595 − 1.92i)11-s + (1.90 − 0.624i)12-s + (5.27 + 1.20i)13-s + (3.52 + 1.26i)14-s + (3.72 − 0.849i)15-s + (2.37 − 3.21i)16-s + (−0.391 + 0.574i)17-s + ⋯
L(s)  = 1  + (−0.972 + 0.231i)2-s + (0.570 + 0.0860i)3-s + (0.892 − 0.450i)4-s + (1.58 − 0.623i)5-s + (−0.575 + 0.0483i)6-s + (−0.837 − 0.546i)7-s + (−0.764 + 0.644i)8-s + (0.318 + 0.0982i)9-s + (−1.40 + 0.974i)10-s + (−0.179 − 0.581i)11-s + (0.548 − 0.180i)12-s + (1.46 + 0.333i)13-s + (0.941 + 0.337i)14-s + (0.960 − 0.219i)15-s + (0.594 − 0.804i)16-s + (−0.0949 + 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.874 + 0.485i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.874 + 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39813 - 0.362430i\)
\(L(\frac12)\) \(\approx\) \(1.39813 - 0.362430i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.327i)T \)
3 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (2.21 + 1.44i)T \)
good5 \( 1 + (-3.55 + 1.39i)T + (3.66 - 3.40i)T^{2} \)
11 \( 1 + (0.595 + 1.92i)T + (-9.08 + 6.19i)T^{2} \)
13 \( 1 + (-5.27 - 1.20i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (0.391 - 0.574i)T + (-6.21 - 15.8i)T^{2} \)
19 \( 1 + (-0.251 + 0.436i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.78 + 2.62i)T + (-8.40 + 21.4i)T^{2} \)
29 \( 1 + (7.98 - 3.84i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-1.38 - 2.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.466 + 6.23i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (5.04 + 4.02i)T + (9.12 + 39.9i)T^{2} \)
43 \( 1 + (-3.66 + 2.91i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-8.78 - 8.14i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (0.582 + 7.77i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (1.58 - 4.03i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (-0.235 - 0.0176i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (-4.38 + 2.53i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.33 - 8.99i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (-6.25 - 6.73i)T + (-5.45 + 72.7i)T^{2} \)
79 \( 1 + (-5.83 - 3.36i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.91 + 12.7i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (1.79 - 5.82i)T + (-73.5 - 50.1i)T^{2} \)
97 \( 1 - 18.2iT - 97T^{2} \)
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

−10.46650971014452735926363636454, −9.451700680257313155970699580594, −9.063116510314425727227234991498, −8.318701906195392896226945916033, −7.03096822570766625260825338692, −6.17758183974267371306266970312, −5.50700981681189536902426053031, −3.70673227244885760565276301680, −2.33380489876074462604797656347, −1.15209176589911674962237354271, 1.68115675183744617953146145102, 2.58832652707073704012939725466, 3.54343329922594467948703344244, 5.77668311705559119909419948132, 6.28994483573397140489536115001, 7.24741759515157728453987788832, 8.359810737966742119581818053044, 9.298938739239892155830541252970, 9.722087392909814021155721402232, 10.41395422275556172391537021847

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