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Properties

Label 2-588-28.19-c1-0-14
Degree $2$
Conductor $588$
Sign $0.427 - 0.904i$
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  + (−1.04 + 0.956i)2-s + (0.5 + 0.866i)3-s + (0.171 − 1.99i)4-s + (0.110 + 0.0639i)5-s + (−1.34 − 0.424i)6-s + (1.72 + 2.24i)8-s + (−0.499 + 0.866i)9-s + (−0.176 + 0.0392i)10-s + (3.45 − 1.99i)11-s + (1.81 − 0.847i)12-s − 0.891i·13-s + 0.127i·15-s + (−3.94 − 0.685i)16-s + (5.04 − 2.91i)17-s + (−0.306 − 1.38i)18-s + (−3.15 + 5.46i)19-s + ⋯
L(s)  = 1  + (−0.736 + 0.676i)2-s + (0.288 + 0.499i)3-s + (0.0859 − 0.996i)4-s + (0.0495 + 0.0286i)5-s + (−0.550 − 0.173i)6-s + (0.610 + 0.792i)8-s + (−0.166 + 0.288i)9-s + (−0.0558 + 0.0124i)10-s + (1.04 − 0.602i)11-s + (0.522 − 0.244i)12-s − 0.247i·13-s + 0.0330i·15-s + (−0.985 − 0.171i)16-s + (1.22 − 0.706i)17-s + (−0.0723 − 0.325i)18-s + (−0.724 + 1.25i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.427 - 0.904i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.427 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02661 + 0.650520i\)
\(L(\frac12)\) \(\approx\) \(1.02661 + 0.650520i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.04 - 0.956i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-0.110 - 0.0639i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.45 + 1.99i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.891iT - 13T^{2} \)
17 \( 1 + (-5.04 + 2.91i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.15 - 5.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.72 - 3.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + (-4.22 - 7.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.33 + 7.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.24iT - 41T^{2} \)
43 \( 1 + 0.881iT - 43T^{2} \)
47 \( 1 + (5.00 - 8.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.76 - 6.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.294 + 0.509i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.46 - 0.843i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.50 - 3.17i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 + (13.4 - 7.75i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.50 - 4.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.48T + 83T^{2} \)
89 \( 1 + (0.389 + 0.224i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.2iT - 97T^{2} \)
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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

−10.53426725654996697699421688134, −9.837934329833412660700815083881, −9.063631134651854426557443723941, −8.293694043694521128118050464170, −7.46759573733062290988556624725, −6.33810427178643087045717777039, −5.57049916735520013694210209184, −4.38339493930566932493631829286, −3.05207718231279870285169070007, −1.24642546959070933782133239679, 1.10199743558070483495611886865, 2.32510112290919404060774869041, 3.53586498627527150619712828640, 4.64855012114582439737574944890, 6.38556550068643938905961327355, 7.09316101076100770686608690811, 8.070495769195466544140753677215, 8.879169640199512797356176677059, 9.596693646273019460406236547081, 10.43786289916027124895742216364

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