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Properties

Label 2-588-84.23-c1-0-43
Degree $2$
Conductor $588$
Sign $0.967 - 0.252i$
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  + (1.36 + 0.382i)2-s + (−0.683 − 1.59i)3-s + (1.70 + 1.04i)4-s + (1.97 + 1.13i)5-s + (−0.323 − 2.42i)6-s + (1.92 + 2.06i)8-s + (−2.06 + 2.17i)9-s + (2.25 + 2.30i)10-s + (2.15 + 3.73i)11-s + (0.488 − 3.42i)12-s − 0.406·13-s + (0.463 − 3.91i)15-s + (1.83 + 3.55i)16-s + (3.73 − 2.15i)17-s + (−3.64 + 2.17i)18-s + (−4.70 − 2.71i)19-s + ⋯
L(s)  = 1  + (0.962 + 0.270i)2-s + (−0.394 − 0.918i)3-s + (0.853 + 0.520i)4-s + (0.882 + 0.509i)5-s + (−0.131 − 0.991i)6-s + (0.681 + 0.731i)8-s + (−0.688 + 0.725i)9-s + (0.711 + 0.728i)10-s + (0.651 + 1.12i)11-s + (0.140 − 0.990i)12-s − 0.112·13-s + (0.119 − 1.01i)15-s + (0.458 + 0.888i)16-s + (0.907 − 0.523i)17-s + (−0.858 + 0.512i)18-s + (−1.07 − 0.622i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.967 - 0.252i$
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: \((2,\ 588,\ (\ :1/2),\ 0.967 - 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68056 + 0.344245i\)
\(L(\frac12)\) \(\approx\) \(2.68056 + 0.344245i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.382i)T \)
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

−10.92477457978192622435991160941, −10.10956935934786519828271236355, −8.861040259001261909410338153745, −7.52504623536937448980702037731, −6.98851395127847212638448290093, −6.18390748426350988949462070820, −5.42271020988015676267891917656, −4.28596343376644111783710893257, −2.67815193608538514165517533635, −1.83625116722310617567644306552, 1.44588099033128949341128608360, 3.14995274560318805007018331922, 4.04749160694134297361673342353, 5.12681344307402803940256955379, 5.89897292345080101194068627480, 6.40724836825995999792500205631, 8.118374702404168855767244817562, 9.259196112490317577627495210398, 9.910362438053084292503737491256, 10.82280403111620600657312774420

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