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Properties

Label 2-588-28.19-c1-0-13
Degree $2$
Conductor $588$
Sign $0.945 - 0.327i$
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  + (−0.546 − 1.30i)2-s + (0.5 + 0.866i)3-s + (−1.40 + 1.42i)4-s + (3.33 + 1.92i)5-s + (0.856 − 1.12i)6-s + (2.62 + 1.05i)8-s + (−0.499 + 0.866i)9-s + (0.690 − 5.40i)10-s + (1.17 − 0.681i)11-s + (−1.93 − 0.502i)12-s + 0.369i·13-s + 3.85i·15-s + (−0.0640 − 3.99i)16-s + (−3.89 + 2.25i)17-s + (1.40 + 0.178i)18-s + (−0.0330 + 0.0573i)19-s + ⋯
L(s)  = 1  + (−0.386 − 0.922i)2-s + (0.288 + 0.499i)3-s + (−0.701 + 0.712i)4-s + (1.49 + 0.862i)5-s + (0.349 − 0.459i)6-s + (0.928 + 0.371i)8-s + (−0.166 + 0.288i)9-s + (0.218 − 1.71i)10-s + (0.355 − 0.205i)11-s + (−0.558 − 0.144i)12-s + 0.102i·13-s + 0.995i·15-s + (−0.0160 − 0.999i)16-s + (−0.945 + 0.545i)17-s + (0.330 + 0.0421i)18-s + (−0.00759 + 0.0131i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.945 - 0.327i$
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: \((2,\ 588,\ (\ :1/2),\ 0.945 - 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55079 + 0.260766i\)
\(L(\frac12)\) \(\approx\) \(1.55079 + 0.260766i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.546 + 1.30i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-3.33 - 1.92i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.17 + 0.681i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.369iT - 13T^{2} \)
17 \( 1 + (3.89 - 2.25i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0330 - 0.0573i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.77 - 1.60i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.11T + 29T^{2} \)
31 \( 1 + (-3.01 - 5.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.74 + 4.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.45iT - 41T^{2} \)
43 \( 1 - 6.30iT - 43T^{2} \)
47 \( 1 + (-0.712 + 1.23i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.27 - 2.20i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.71 + 2.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.23 + 0.715i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.45 + 4.88i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 + (-1.56 + 0.900i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.8 - 6.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + (1.11 + 0.646i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.88iT - 97T^{2} \)
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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

−10.84024206003992497931093401895, −9.841688101856048559098312013383, −9.300903474049887408719014401018, −8.552682469261715221159229245091, −7.21447197524891685674187914908, −6.17570168477765418321585930140, −5.03531303776239787150334405432, −3.75301329299970449565869269876, −2.70597390571054568658277664590, −1.76541714830115424529770721974, 1.08189584036833152324603981280, 2.31154690177441984756740923904, 4.43412569599311579033697404616, 5.34789594934479801126661271181, 6.24180894675941582929097733764, 6.92462179396700704340273160512, 8.107410153426268010814845490185, 8.929327437226075434231534665746, 9.463507634459733119691405745565, 10.19316838926093478287972080172

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