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Properties

Label 2-588-84.11-c1-0-27
Degree $2$
Conductor $588$
Sign $0.873 - 0.487i$
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  + (−1.01 + 0.988i)2-s + (−1.03 + 1.38i)3-s + (0.0474 − 1.99i)4-s + (−1.97 + 1.13i)5-s + (−0.323 − 2.42i)6-s + (1.92 + 2.06i)8-s + (−0.852 − 2.87i)9-s + (0.870 − 3.10i)10-s + (2.15 − 3.73i)11-s + (2.72 + 2.13i)12-s − 0.406·13-s + (0.463 − 3.91i)15-s + (−3.99 − 0.189i)16-s + (−3.73 − 2.15i)17-s + (3.70 + 2.06i)18-s + (4.70 − 2.71i)19-s + ⋯
L(s)  = 1  + (−0.715 + 0.698i)2-s + (−0.598 + 0.801i)3-s + (0.0237 − 0.999i)4-s + (−0.882 + 0.509i)5-s + (−0.131 − 0.991i)6-s + (0.681 + 0.731i)8-s + (−0.284 − 0.958i)9-s + (0.275 − 0.980i)10-s + (0.651 − 1.12i)11-s + (0.786 + 0.617i)12-s − 0.112·13-s + (0.119 − 1.01i)15-s + (−0.998 − 0.0474i)16-s + (−0.907 − 0.523i)17-s + (0.873 + 0.487i)18-s + (1.07 − 0.622i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.873 - 0.487i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.873 - 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.586229 + 0.152485i\)
\(L(\frac12)\) \(\approx\) \(0.586229 + 0.152485i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.01 - 0.988i)T \)
3 \( 1 + (1.03 - 1.38i)T \)
7 \( 1 \)
good5 \( 1 + (1.97 - 1.13i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.15 + 3.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.406T + 13T^{2} \)
17 \( 1 + (3.73 + 2.15i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.70 + 2.71i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.581 - 1.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.72iT - 29T^{2} \)
31 \( 1 + (-5.05 - 2.91i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.91 - 6.78i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.56iT - 41T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + (-1.16 - 2.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.74 - 2.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.95 + 3.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.22 + 9.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.78 - 3.91i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.88T + 71T^{2} \)
73 \( 1 + (1.40 - 2.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.46 + 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.641T + 83T^{2} \)
89 \( 1 + (-12.1 + 6.99i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.79019548731503379071125987375, −9.776884805968559869167077345797, −9.050905982766368355639943878942, −8.198997256087969917593294055445, −7.08495128992879238518654128544, −6.39569050346078535099387604346, −5.35879470098719210137632576700, −4.34321348512774829969276528507, −3.12687366431283183132065507282, −0.61794803405078768720550515439, 1.02063657989928198401461179846, 2.27657263162120635469407837501, 3.92920536395194487639215727653, 4.80575139081298122113509749630, 6.37877492516458776611850284619, 7.34301995351777854179089936044, 7.916327268295802509083983492781, 8.842531945477282559721369328638, 9.800410423774457318148163046559, 10.78716630490310123475991222011

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