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Properties

Label 2-588-196.55-c1-0-2
Degree $2$
Conductor $588$
Sign $-0.412 - 0.910i$
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  + (−0.611 − 1.27i)2-s + (0.900 + 0.433i)3-s + (−1.25 + 1.55i)4-s + (−0.776 + 1.61i)5-s + (0.00226 − 1.41i)6-s + (−0.448 + 2.60i)7-s + (2.75 + 0.642i)8-s + (0.623 + 0.781i)9-s + (2.53 + 0.00405i)10-s + (−3.75 − 2.99i)11-s + (−1.80 + 0.861i)12-s + (0.836 + 0.667i)13-s + (3.59 − 1.02i)14-s + (−1.39 + 1.11i)15-s + (−0.865 − 3.90i)16-s + (−5.54 − 1.26i)17-s + ⋯
L(s)  = 1  + (−0.432 − 0.901i)2-s + (0.520 + 0.250i)3-s + (−0.625 + 0.779i)4-s + (−0.347 + 0.721i)5-s + (0.000924 − 0.577i)6-s + (−0.169 + 0.985i)7-s + (0.973 + 0.227i)8-s + (0.207 + 0.260i)9-s + (0.800 + 0.00128i)10-s + (−1.13 − 0.902i)11-s + (−0.520 + 0.248i)12-s + (0.231 + 0.184i)13-s + (0.961 − 0.273i)14-s + (−0.361 + 0.288i)15-s + (−0.216 − 0.976i)16-s + (−1.34 − 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.412 - 0.910i$
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: \((2,\ 588,\ (\ :1/2),\ -0.412 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.289224 + 0.448526i\)
\(L(\frac12)\) \(\approx\) \(0.289224 + 0.448526i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.611 + 1.27i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (0.448 - 2.60i)T \)
good5 \( 1 + (0.776 - 1.61i)T + (-3.11 - 3.90i)T^{2} \)
11 \( 1 + (3.75 + 2.99i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.836 - 0.667i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (5.54 + 1.26i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + 6.65T + 19T^{2} \)
23 \( 1 + (0.388 - 0.0886i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (1.41 - 6.18i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 1.92T + 31T^{2} \)
37 \( 1 + (-0.0915 + 0.401i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-2.82 + 5.85i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (-5.15 - 10.7i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.04 + 1.31i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (0.839 + 3.67i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-7.27 + 3.50i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (2.37 + 0.543i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 + (-1.61 + 0.368i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (7.88 - 6.29i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 7.64iT - 79T^{2} \)
83 \( 1 + (-3.85 - 4.83i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-6.71 + 5.35i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + 2.84iT - 97T^{2} \)
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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

−10.99756050948784563887901706302, −10.28314873551028681794019037071, −9.026977195371972850626773675861, −8.701761452847259456644841913115, −7.76736320160082026769959780747, −6.62074388963217142689026684432, −5.23962134264089899529839536415, −3.99922920591307928181470832077, −2.92390956421091652485502573649, −2.26115026707542212261568521561, 0.30564982615391446726386670419, 2.08528761357443216313568615314, 4.16892146634564864136058393935, 4.64304413779865881697029976788, 6.09988046589524633559532202972, 7.04335453641304969519927455983, 7.81236501857866590097364571781, 8.463509170697664846034805272505, 9.283836545330774767409247438591, 10.32748311847868768161380866365

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