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Properties

Label 2-592-37.30-c1-0-9
Degree $2$
Conductor $592$
Sign $0.957 + 0.290i$
Analytic cond. $4.72714$
Root an. cond. $2.17419$
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  + (−0.326 − 0.118i)3-s + (2.57 − 0.453i)5-s + (0.361 + 2.04i)7-s + (−2.20 − 1.85i)9-s + (2.99 − 5.19i)11-s + (2.64 + 3.15i)13-s + (−0.893 − 0.157i)15-s + (−0.618 + 0.737i)17-s + (−0.534 + 1.46i)19-s + (0.125 − 0.711i)21-s + (5.51 − 3.18i)23-s + (1.70 − 0.622i)25-s + (1.02 + 1.76i)27-s + (−3.51 − 2.02i)29-s + 3.39i·31-s + ⋯
L(s)  = 1  + (−0.188 − 0.0685i)3-s + (1.15 − 0.202i)5-s + (0.136 + 0.773i)7-s + (−0.735 − 0.616i)9-s + (0.903 − 1.56i)11-s + (0.733 + 0.874i)13-s + (−0.230 − 0.0406i)15-s + (−0.150 + 0.178i)17-s + (−0.122 + 0.337i)19-s + (0.0273 − 0.155i)21-s + (1.15 − 0.664i)23-s + (0.341 − 0.124i)25-s + (0.196 + 0.340i)27-s + (−0.652 − 0.376i)29-s + 0.610i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.957 + 0.290i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.957 + 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71251 - 0.253846i\)
\(L(\frac12)\) \(\approx\) \(1.71251 - 0.253846i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (-6.07 + 0.383i)T \)
good3 \( 1 + (0.326 + 0.118i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (-2.57 + 0.453i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.361 - 2.04i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.99 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.64 - 3.15i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.618 - 0.737i)T + (-2.95 - 16.7i)T^{2} \)
19 \( 1 + (0.534 - 1.46i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (-5.51 + 3.18i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.51 + 2.02i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.39iT - 31T^{2} \)
41 \( 1 + (-7.94 + 6.66i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 3.76iT - 43T^{2} \)
47 \( 1 + (3.08 + 5.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.39 - 7.90i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (5.02 + 0.885i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (6.25 + 7.45i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-1.83 - 10.3i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-10.1 - 3.69i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 3.55T + 73T^{2} \)
79 \( 1 + (2.51 - 0.442i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (5.29 + 4.44i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (16.0 + 2.82i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (14.1 - 8.15i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.91140631937144956590131968805, −9.409216714114800989183751267806, −9.009686210729789372327575794814, −8.391535296462959562275225824777, −6.62749621166871186306593261026, −6.03371921041104512609260836342, −5.49005987784447623217352740924, −3.91417852240324201379100928908, −2.67321274653307903760595366080, −1.24308338117462839674999484427, 1.45535404078502225491878924476, 2.73425807039215330457893586129, 4.22378172614259038102744875369, 5.25009968191743513687641241382, 6.15602639837260577372367680624, 7.08748203872021635076315574957, 7.983358153713834916790107788626, 9.304592169584784186347190162004, 9.750389812164428603544339071558, 10.84077840518373280644110222110

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