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Properties

Label 2-588-196.27-c1-0-19
Degree $2$
Conductor $588$
Sign $0.775 + 0.630i$
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  + (−1.25 − 0.643i)2-s + (0.222 − 0.974i)3-s + (1.17 + 1.62i)4-s + (0.631 + 0.144i)5-s + (−0.907 + 1.08i)6-s + (−2.32 + 1.26i)7-s + (−0.434 − 2.79i)8-s + (−0.900 − 0.433i)9-s + (−0.703 − 0.588i)10-s + (2.50 + 5.19i)11-s + (1.84 − 0.782i)12-s + (−2.15 − 4.47i)13-s + (3.74 − 0.102i)14-s + (0.281 − 0.584i)15-s + (−1.25 + 3.79i)16-s + (6.08 − 4.84i)17-s + ⋯
L(s)  = 1  + (−0.890 − 0.454i)2-s + (0.128 − 0.562i)3-s + (0.586 + 0.810i)4-s + (0.282 + 0.0645i)5-s + (−0.370 + 0.442i)6-s + (−0.877 + 0.479i)7-s + (−0.153 − 0.988i)8-s + (−0.300 − 0.144i)9-s + (−0.222 − 0.186i)10-s + (0.754 + 1.56i)11-s + (0.531 − 0.225i)12-s + (−0.598 − 1.24i)13-s + (0.999 − 0.0273i)14-s + (0.0726 − 0.150i)15-s + (−0.312 + 0.949i)16-s + (1.47 − 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.775 + 0.630i$
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: \((2,\ 588,\ (\ :1/2),\ 0.775 + 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.970982 - 0.344851i\)
\(L(\frac12)\) \(\approx\) \(0.970982 - 0.344851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 + 0.643i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (2.32 - 1.26i)T \)
good5 \( 1 + (-0.631 - 0.144i)T + (4.50 + 2.16i)T^{2} \)
11 \( 1 + (-2.50 - 5.19i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (2.15 + 4.47i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (-6.08 + 4.84i)T + (3.78 - 16.5i)T^{2} \)
19 \( 1 - 5.84T + 19T^{2} \)
23 \( 1 + (-2.05 - 1.64i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (-4.35 - 5.46i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 4.28T + 31T^{2} \)
37 \( 1 + (-1.37 - 1.72i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-7.63 - 1.74i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.773 + 0.176i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (0.374 - 0.180i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-2.35 + 2.95i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (1.21 + 5.32i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (3.31 - 2.64i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + (3.04 + 2.42i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (2.91 - 6.05i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + 2.62iT - 79T^{2} \)
83 \( 1 + (-5.45 - 2.62i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-3.02 + 6.28i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 - 2.58iT - 97T^{2} \)
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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

−10.15430424352879898784059685888, −9.757725561483422414237771092002, −9.163586479994357968877425778758, −7.76365983036909100237037181470, −7.34601194434902356032997982259, −6.36608643930829963169632692946, −5.14279494704190865962887506161, −3.30930595120400625931088032941, −2.57062341158149893183415501535, −1.05996602038489807953139869971, 1.06846100148440019765285071936, 2.94473354450167219900903319410, 4.09294942324127546398182296070, 5.69718523830017671184514072722, 6.20749516229960874652021955313, 7.31387393699136726920640557659, 8.274401580093200129905879051697, 9.241743767581838032109657434101, 9.712150706531647599655139122585, 10.45622033734792031366697322189

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