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Properties

Label 2-588-84.23-c1-0-65
Degree $2$
Conductor $588$
Sign $-0.775 + 0.631i$
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 + (−0.683 − 1.59i)3-s + (0.0474 + 1.99i)4-s + (−1.97 − 1.13i)5-s + (0.880 − 2.28i)6-s + (−1.92 + 2.06i)8-s + (−2.06 + 2.17i)9-s + (−0.870 − 3.10i)10-s + (−2.15 − 3.73i)11-s + (3.14 − 1.44i)12-s + 0.406·13-s + (−0.463 + 3.91i)15-s + (−3.99 + 0.189i)16-s + (−3.73 + 2.15i)17-s + (−4.23 + 0.162i)18-s + (−4.70 − 2.71i)19-s + ⋯
L(s)  = 1  + (0.715 + 0.698i)2-s + (−0.394 − 0.918i)3-s + (0.0237 + 0.999i)4-s + (−0.882 − 0.509i)5-s + (0.359 − 0.933i)6-s + (−0.681 + 0.731i)8-s + (−0.688 + 0.725i)9-s + (−0.275 − 0.980i)10-s + (−0.651 − 1.12i)11-s + (0.909 − 0.416i)12-s + 0.112·13-s + (−0.119 + 1.01i)15-s + (−0.998 + 0.0474i)16-s + (−0.907 + 0.523i)17-s + (−0.999 + 0.0382i)18-s + (−1.07 − 0.622i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.775 + 0.631i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124421 - 0.349989i\)
\(L(\frac12)\) \(\approx\) \(0.124421 - 0.349989i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.683 + 1.59i)T \)
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

−10.99136034932757104345033297996, −8.926543618309772526152445611516, −8.245344452396893026838007788635, −7.69478384383790179833554113740, −6.60682098539544630800613458482, −5.89400822840324746413653123673, −4.86976279338651441999985024078, −3.84166874106588999702073471621, −2.44363501426904513533650991322, −0.16062118560939502071029930788, 2.33965768821184562664155945161, 3.58576585931371197522615593302, 4.35521269609977747768750744803, 5.14370250809815133078173424798, 6.31993729782137578001341096346, 7.27213663034111815808473315756, 8.640389797454423515050417912929, 9.681942149395589212686110789870, 10.42664198199682420924475798945, 11.04160151263024778174096182498

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