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Properties

Label 2-588-196.27-c1-0-12
Degree $2$
Conductor $588$
Sign $0.426 - 0.904i$
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  + (−0.0968 + 1.41i)2-s + (0.222 − 0.974i)3-s + (−1.98 − 0.273i)4-s + (−3.84 − 0.877i)5-s + (1.35 + 0.408i)6-s + (2.22 + 1.42i)7-s + (0.577 − 2.76i)8-s + (−0.900 − 0.433i)9-s + (1.61 − 5.34i)10-s + (0.211 + 0.438i)11-s + (−0.707 + 1.87i)12-s + (1.80 + 3.74i)13-s + (−2.22 + 3.00i)14-s + (−1.71 + 3.55i)15-s + (3.85 + 1.08i)16-s + (3.87 − 3.09i)17-s + ⋯
L(s)  = 1  + (−0.0684 + 0.997i)2-s + (0.128 − 0.562i)3-s + (−0.990 − 0.136i)4-s + (−1.72 − 0.392i)5-s + (0.552 + 0.166i)6-s + (0.842 + 0.538i)7-s + (0.204 − 0.978i)8-s + (−0.300 − 0.144i)9-s + (0.509 − 1.68i)10-s + (0.0636 + 0.132i)11-s + (−0.204 + 0.540i)12-s + (0.500 + 1.03i)13-s + (−0.594 + 0.803i)14-s + (−0.441 + 0.917i)15-s + (0.962 + 0.270i)16-s + (0.940 − 0.750i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.426 - 0.904i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.426 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.893887 + 0.567027i\)
\(L(\frac12)\) \(\approx\) \(0.893887 + 0.567027i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0968 - 1.41i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (-2.22 - 1.42i)T \)
good5 \( 1 + (3.84 + 0.877i)T + (4.50 + 2.16i)T^{2} \)
11 \( 1 + (-0.211 - 0.438i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.80 - 3.74i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (-3.87 + 3.09i)T + (3.78 - 16.5i)T^{2} \)
19 \( 1 - 1.26T + 19T^{2} \)
23 \( 1 + (-1.01 - 0.811i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (-3.84 - 4.82i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 + (-5.94 - 7.45i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (0.785 + 0.179i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (9.95 - 2.27i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (-7.29 + 3.51i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-4.02 + 5.04i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (0.246 + 1.08i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-4.52 + 3.61i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + 5.76iT - 67T^{2} \)
71 \( 1 + (3.14 + 2.50i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (-6.35 + 13.1i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 - 17.2iT - 79T^{2} \)
83 \( 1 + (4.95 + 2.38i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (6.19 - 12.8i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + 3.59iT - 97T^{2} \)
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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

−11.13244056411885516900204907250, −9.587042841539538657139570394380, −8.568093121981987451374680679353, −8.204419522501501671780217099643, −7.38453922577701981598784841425, −6.65038846724976135051497944023, −5.23970088535007853046681589113, −4.51655373402569203759112972101, −3.38129547380424135330732559325, −1.11109124204195852244556357175, 0.832081180077253134809643971833, 2.92287883738868295241326049320, 3.82494427268479375049193085504, 4.38365275527071830371832703794, 5.58984831099501925926449086491, 7.44584634000412427464082698921, 8.111196104325607402868123759910, 8.613916353948403117363212079023, 10.09462899315479972625725769388, 10.62804404514870004658746130113

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