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Properties

Label 2-588-28.3-c1-0-20
Degree $2$
Conductor $588$
Sign $0.382 + 0.923i$
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  + (−0.00239 − 1.41i)2-s + (−0.5 + 0.866i)3-s + (−1.99 + 0.00677i)4-s + (3.15 − 1.82i)5-s + (1.22 + 0.705i)6-s + (0.0143 + 2.82i)8-s + (−0.499 − 0.866i)9-s + (−2.58 − 4.46i)10-s + (5.25 + 3.03i)11-s + (0.994 − 1.73i)12-s − 0.483i·13-s + 3.64i·15-s + (3.99 − 0.0271i)16-s + (−2.21 − 1.27i)17-s + (−1.22 + 0.709i)18-s + (0.609 + 1.05i)19-s + ⋯
L(s)  = 1  + (−0.00169 − 0.999i)2-s + (−0.288 + 0.499i)3-s + (−0.999 + 0.00338i)4-s + (1.41 − 0.815i)5-s + (0.500 + 0.287i)6-s + (0.00508 + 0.999i)8-s + (−0.166 − 0.288i)9-s + (−0.818 − 1.41i)10-s + (1.58 + 0.915i)11-s + (0.286 − 0.500i)12-s − 0.134i·13-s + 0.941i·15-s + (0.999 − 0.00677i)16-s + (−0.536 − 0.309i)17-s + (−0.288 + 0.167i)18-s + (0.139 + 0.242i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.382 + 0.923i$
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: \((2,\ 588,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34617 - 0.899764i\)
\(L(\frac12)\) \(\approx\) \(1.34617 - 0.899764i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.00239 + 1.41i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-3.15 + 1.82i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.25 - 3.03i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.483iT - 13T^{2} \)
17 \( 1 + (2.21 + 1.27i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.609 - 1.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.00 - 1.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.21T + 29T^{2} \)
31 \( 1 + (-3.15 + 5.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.595 + 1.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.59iT - 41T^{2} \)
43 \( 1 - 3.51iT - 43T^{2} \)
47 \( 1 + (5.83 + 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.31 - 2.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.580 - 1.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.180 - 0.104i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.53 - 0.888i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.13iT - 71T^{2} \)
73 \( 1 + (-5.23 - 3.02i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.75 - 1.58i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.49T + 83T^{2} \)
89 \( 1 + (10.6 - 6.14i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.8iT - 97T^{2} \)
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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

−10.24758458129088799313171902629, −9.758395280040978094208883714321, −9.208923482136806184739960574204, −8.404373064890181979648004775580, −6.65483474253239458054302817754, −5.67353162371071816352655173295, −4.75257884255324862120208329617, −3.96881571118704783888946907784, −2.31480514341231731411221734495, −1.23430459488667104975079958474, 1.39371064455537622487571646847, 3.08600168319826831350357650559, 4.57517245927053292575456610635, 5.83602794757191129896211487143, 6.52211069971746047935375761246, 6.70950261839050080746623306966, 8.195919284599835782058624519379, 9.010602809461528280006293261986, 9.830490006982062598250497616843, 10.68983720899062545103042291338

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