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Properties

Label 2-588-28.3-c1-0-35
Degree $2$
Conductor $588$
Sign $-0.870 - 0.491i$
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  + (−0.306 − 1.38i)2-s + (0.5 − 0.866i)3-s + (−1.81 + 0.847i)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 + (0.122 + 0.133i)10-s + (−3.45 − 1.99i)11-s + (−0.171 + 1.99i)12-s − 0.891i·13-s + 0.127i·15-s + (2.56 − 3.07i)16-s + (−5.04 − 2.91i)17-s + (−1.04 + 0.956i)18-s + (−3.15 − 5.46i)19-s + ⋯
L(s)  = 1  + (−0.217 − 0.976i)2-s + (0.288 − 0.499i)3-s + (−0.905 + 0.423i)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 + (0.0386 + 0.0421i)10-s + (−1.04 − 0.602i)11-s + (−0.0496 + 0.575i)12-s − 0.247i·13-s + 0.0330i·15-s + (0.640 − 0.767i)16-s + (−1.22 − 0.706i)17-s + (−0.245 + 0.225i)18-s + (−0.724 − 1.25i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.870 - 0.491i$
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: \((2,\ 588,\ (\ :1/2),\ -0.870 - 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.157873 + 0.600919i\)
\(L(\frac12)\) \(\approx\) \(0.157873 + 0.600919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.306 + 1.38i)T \)
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

−10.18973307135694456009293397339, −9.417075953928102245940741661130, −8.410937424171932491976788108736, −7.888625231989284967080599346772, −6.68799657789272395685694454277, −5.37373953802764084238386776792, −4.29222705905520413704696195415, −2.98349459836265692189367659183, −2.16086967038565974041988473684, −0.34292690022527881172801830591, 2.20047225593198241109745707993, 4.04101765242932016427194346036, 4.64374969535583887868921449383, 5.87785606901487067429973943989, 6.65505935015222619848522723546, 7.961105051370456532761385554365, 8.305966639426005079608816280929, 9.341625929642543825446297870252, 10.31609512544999623695734918766, 10.63546395348270099204484896313

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