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Properties

Label 2-588-28.3-c1-0-30
Degree $2$
Conductor $588$
Sign $0.978 - 0.204i$
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  + (1.40 + 0.178i)2-s + (−0.5 + 0.866i)3-s + (1.93 + 0.502i)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 + (5.02 − 2.10i)10-s + (−1.17 − 0.681i)11-s + (−1.40 + 1.42i)12-s − 0.369i·13-s + 3.85i·15-s + (3.49 + 1.94i)16-s + (−3.89 − 2.25i)17-s + (−0.546 − 1.30i)18-s + (0.0330 + 0.0573i)19-s + ⋯
L(s)  = 1  + (0.991 + 0.126i)2-s + (−0.288 + 0.499i)3-s + (0.967 + 0.251i)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 + (1.59 − 0.666i)10-s + (−0.355 − 0.205i)11-s + (−0.404 + 0.411i)12-s − 0.102i·13-s + 0.995i·15-s + (0.873 + 0.486i)16-s + (−0.945 − 0.545i)17-s + (−0.128 − 0.307i)18-s + (0.00759 + 0.0131i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.978 - 0.204i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.978 - 0.204i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.00383 + 0.311150i\)
\(L(\frac12)\) \(\approx\) \(3.00383 + 0.311150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.178i)T \)
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

−10.79013782258797823703294286241, −9.913562697810440139684862902248, −9.151823261972492412766935119751, −8.070083881951520251794444294481, −6.67844294180859371821897595476, −5.90400739364354938885172381131, −5.15417126540930412512252404090, −4.46765348884984083116591223981, −2.96366112504726810216236456762, −1.70259850296755762347576629265, 1.91983390417861518422638830240, 2.51352636906772470016397190993, 4.02258909520958459985856958264, 5.42616884030649192305168965916, 5.99383033920057806862987160782, 6.77138704957943074055057601222, 7.54810057540467837639675015228, 9.082799768755552801803355771232, 10.20718307420885897659141470131, 10.70204681966942387924023881359

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