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Properties

Label 2-588-196.55-c1-0-0
Degree $2$
Conductor $588$
Sign $-0.391 + 0.920i$
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  + (0.131 + 1.40i)2-s + (−0.900 − 0.433i)3-s + (−1.96 + 0.369i)4-s + (−1.40 + 2.91i)5-s + (0.492 − 1.32i)6-s + (−0.602 + 2.57i)7-s + (−0.777 − 2.71i)8-s + (0.623 + 0.781i)9-s + (−4.29 − 1.59i)10-s + (4.48 + 3.57i)11-s + (1.93 + 0.520i)12-s + (−5.18 − 4.13i)13-s + (−3.70 − 0.510i)14-s + (2.53 − 2.01i)15-s + (3.72 − 1.45i)16-s + (−2.33 − 0.533i)17-s + ⋯
L(s)  = 1  + (0.0926 + 0.995i)2-s + (−0.520 − 0.250i)3-s + (−0.982 + 0.184i)4-s + (−0.628 + 1.30i)5-s + (0.201 − 0.541i)6-s + (−0.227 + 0.973i)7-s + (−0.274 − 0.961i)8-s + (0.207 + 0.260i)9-s + (−1.35 − 0.504i)10-s + (1.35 + 1.07i)11-s + (0.557 + 0.150i)12-s + (−1.43 − 1.14i)13-s + (−0.990 − 0.136i)14-s + (0.653 − 0.521i)15-s + (0.931 − 0.362i)16-s + (−0.566 − 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.391 + 0.920i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.391 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.229402 - 0.346862i\)
\(L(\frac12)\) \(\approx\) \(0.229402 - 0.346862i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.131 - 1.40i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (0.602 - 2.57i)T \)
good5 \( 1 + (1.40 - 2.91i)T + (-3.11 - 3.90i)T^{2} \)
11 \( 1 + (-4.48 - 3.57i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (5.18 + 4.13i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (2.33 + 0.533i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 - 0.956T + 19T^{2} \)
23 \( 1 + (3.93 - 0.898i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.91 + 8.36i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 1.40T + 31T^{2} \)
37 \( 1 + (1.60 - 7.01i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (0.108 - 0.225i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (2.58 + 5.37i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (0.0885 - 0.111i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (0.185 + 0.811i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-8.03 + 3.87i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (4.45 + 1.01i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 + (3.03 - 0.693i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (11.1 - 8.85i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 9.02iT - 79T^{2} \)
83 \( 1 + (-10.1 - 12.6i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (0.234 - 0.186i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 - 3.75iT - 97T^{2} \)
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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

−11.61887174440540719665897190316, −10.08538549784170081003693481049, −9.703890423987418099684128598393, −8.366473811697095242971693695558, −7.42294044233935840693161748198, −6.85365133993741591043238010622, −6.09891412983541001576342710844, −5.02508985137426721979601667078, −3.90733593183774366017190248113, −2.55235124600367747906498712975, 0.25372121952082535825343899839, 1.49440361358050822158458748329, 3.57298369294653111828307444998, 4.33705196334579334043181909657, 4.92650188332089084118964313825, 6.30840938314865750073319858447, 7.51530296499986671022196921527, 8.859957116806621042832331816311, 9.162686124690348049945628210906, 10.20397351007830711544118963696

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