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Properties

Label 2-588-147.5-c1-0-11
Degree $2$
Conductor $588$
Sign $0.764 + 0.644i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.975 + 1.43i)3-s + (−2.23 − 1.41i)7-s + (−1.09 − 2.79i)9-s + (5.35 − 4.27i)13-s + (−3.99 − 2.30i)19-s + (4.20 − 1.81i)21-s + (4.94 + 0.745i)25-s + (5.06 + 1.15i)27-s + (9.62 − 5.55i)31-s + (−2.50 − 0.771i)37-s + (0.886 + 11.8i)39-s + (−6.08 − 2.92i)43-s + (2.99 + 6.32i)49-s + (7.20 − 3.47i)57-s + (4.47 − 14.5i)61-s + ⋯
L(s)  = 1  + (−0.563 + 0.826i)3-s + (−0.845 − 0.534i)7-s + (−0.365 − 0.930i)9-s + (1.48 − 1.18i)13-s + (−0.917 − 0.529i)19-s + (0.917 − 0.397i)21-s + (0.988 + 0.149i)25-s + (0.974 + 0.222i)27-s + (1.72 − 0.997i)31-s + (−0.411 − 0.126i)37-s + (0.141 + 1.89i)39-s + (−0.927 − 0.446i)43-s + (0.428 + 0.903i)49-s + (0.954 − 0.459i)57-s + (0.573 − 1.85i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.764 + 0.644i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.764 + 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.917355 - 0.335288i\)
\(L(\frac12)\) \(\approx\) \(0.917355 - 0.335288i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.975 - 1.43i)T \)
7 \( 1 + (2.23 + 1.41i)T \)
good5 \( 1 + (-4.94 - 0.745i)T^{2} \)
11 \( 1 + (8.06 + 7.48i)T^{2} \)
13 \( 1 + (-5.35 + 4.27i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (3.99 + 2.30i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-9.62 + 5.55i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.50 + 0.771i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (6.08 + 2.92i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (-43.7 + 29.8i)T^{2} \)
59 \( 1 + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-4.47 + 14.5i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (1.12 + 1.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-2.52 + 16.7i)T + (-69.7 - 21.5i)T^{2} \)
79 \( 1 + (0.641 - 1.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + 2.30iT - 97T^{2} \)
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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

−10.61366826403066815585751928508, −9.932957346866608414380360932502, −8.955031174140909381064975675956, −8.116194243512140495989151512260, −6.64300618555924102410001355534, −6.13327923059855243970435238088, −4.98404834871201979531900243763, −3.89499161468844992989346798216, −3.05862249007377743979105855422, −0.64860535172667476546892407768, 1.39888155373144070234945186153, 2.79794804723528566991133051525, 4.20907113441499542432788947861, 5.52787425518504365605582607129, 6.54380162270201223330027571628, 6.73019768363958595702674507831, 8.342453256807889173922589451388, 8.764429741705238021715700920327, 10.05464617262768787268203291106, 10.89898766714367124050227695653

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