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Properties

Label 2-588-196.55-c1-0-25
Degree $2$
Conductor $588$
Sign $0.998 - 0.0467i$
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  + (−1.20 + 0.733i)2-s + (−0.900 − 0.433i)3-s + (0.923 − 1.77i)4-s + (−0.0202 + 0.0419i)5-s + (1.40 − 0.136i)6-s + (2.50 − 0.851i)7-s + (0.184 + 2.82i)8-s + (0.623 + 0.781i)9-s + (−0.00635 − 0.0656i)10-s + (4.22 + 3.36i)11-s + (−1.60 + 1.19i)12-s + (−4.09 − 3.26i)13-s + (−2.40 + 2.86i)14-s + (0.0364 − 0.0290i)15-s + (−2.29 − 3.27i)16-s + (−1.81 − 0.414i)17-s + ⋯
L(s)  = 1  + (−0.854 + 0.518i)2-s + (−0.520 − 0.250i)3-s + (0.461 − 0.886i)4-s + (−0.00904 + 0.0187i)5-s + (0.574 − 0.0556i)6-s + (0.946 − 0.321i)7-s + (0.0651 + 0.997i)8-s + (0.207 + 0.260i)9-s + (−0.00200 − 0.0207i)10-s + (1.27 + 1.01i)11-s + (−0.462 + 0.345i)12-s + (−1.13 − 0.906i)13-s + (−0.642 + 0.766i)14-s + (0.00940 − 0.00750i)15-s + (−0.573 − 0.819i)16-s + (−0.440 − 0.100i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.998 - 0.0467i$
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: \((2,\ 588,\ (\ :1/2),\ 0.998 - 0.0467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.935529 + 0.0218710i\)
\(L(\frac12)\) \(\approx\) \(0.935529 + 0.0218710i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 - 0.733i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (-2.50 + 0.851i)T \)
good5 \( 1 + (0.0202 - 0.0419i)T + (-3.11 - 3.90i)T^{2} \)
11 \( 1 + (-4.22 - 3.36i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (4.09 + 3.26i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (1.81 + 0.414i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + 3.64T + 19T^{2} \)
23 \( 1 + (-7.77 + 1.77i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.460 + 2.01i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 6.23T + 31T^{2} \)
37 \( 1 + (-1.79 + 7.84i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (1.75 - 3.63i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (-2.31 - 4.80i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-0.371 + 0.466i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.113 - 0.497i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-3.00 + 1.44i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-12.0 - 2.74i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 9.62iT - 67T^{2} \)
71 \( 1 + (-1.82 + 0.417i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-7.50 + 5.98i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 6.41iT - 79T^{2} \)
83 \( 1 + (-2.99 - 3.75i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (12.8 - 10.2i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + 6.08iT - 97T^{2} \)
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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

−10.69089032238976815367382845160, −9.772312895792671726605999508878, −8.939129443615403712460549495359, −7.912701929664951069005856357922, −7.12984529653460907206039295079, −6.53169900114603786030014420925, −5.18242782160660604326740980741, −4.52895832470917105579275519366, −2.30351219786087903725995370749, −0.989255396925901729427354785804, 1.11951857248613916659341627965, 2.52573889292238245350165369986, 4.02204549480270019162853210160, 4.94383619543300806094641185174, 6.45143987331630466736958263736, 7.06282639227742602906363582702, 8.500654711789712203677812488280, 8.840936612710460431938182181584, 9.838780166637340927092808441751, 10.80823658865919567216966924142

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