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Properties

Label 2-592-37.33-c1-0-7
Degree $2$
Conductor $592$
Sign $0.824 - 0.566i$
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  + (0.238 + 1.35i)3-s + (0.266 − 0.223i)5-s + (0.365 − 0.307i)7-s + (1.04 − 0.378i)9-s + (1.29 − 2.23i)11-s + (4.21 + 1.53i)13-s + (0.365 + 0.307i)15-s + (−1.88 + 0.687i)17-s + (−0.611 − 3.46i)19-s + (0.503 + 0.422i)21-s + (2.60 + 4.51i)23-s + (−0.847 + 4.80i)25-s + (2.82 + 4.89i)27-s + (−0.114 + 0.197i)29-s + 6.74·31-s + ⋯
L(s)  = 1  + (0.137 + 0.782i)3-s + (0.118 − 0.0998i)5-s + (0.138 − 0.116i)7-s + (0.346 − 0.126i)9-s + (0.389 − 0.675i)11-s + (1.16 + 0.425i)13-s + (0.0944 + 0.0792i)15-s + (−0.458 + 0.166i)17-s + (−0.140 − 0.795i)19-s + (0.109 + 0.0921i)21-s + (0.543 + 0.940i)23-s + (−0.169 + 0.961i)25-s + (0.543 + 0.941i)27-s + (−0.0211 + 0.0367i)29-s + 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.824 - 0.566i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.824 - 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67156 + 0.518670i\)
\(L(\frac12)\) \(\approx\) \(1.67156 + 0.518670i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.42 + 5.57i)T \)
good3 \( 1 + (-0.238 - 1.35i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (-0.266 + 0.223i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.365 + 0.307i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-1.29 + 2.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.21 - 1.53i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.88 - 0.687i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (0.611 + 3.46i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (-2.60 - 4.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.114 - 0.197i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.74T + 31T^{2} \)
41 \( 1 + (10.0 + 3.66i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + 8.85T + 43T^{2} \)
47 \( 1 + (-3.72 - 6.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.35 - 2.81i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (3.43 + 2.87i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.35 - 0.491i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-8.06 + 6.76i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.54 + 8.74i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 - 7.18T + 73T^{2} \)
79 \( 1 + (2.70 - 2.27i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.32 + 0.483i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-2.92 - 2.45i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (9.24 + 16.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.97470447725303359408493722305, −9.759803109363007458175151196023, −9.102195196788969062223753616173, −8.411581811925939345281695689917, −7.11176578954379674020124116235, −6.23128455118106342621507046778, −5.08068668509065410216374301872, −4.08095560799538958916578677504, −3.25375808315123964543184466517, −1.40382789047705352198373546388, 1.27650960778263735689112964499, 2.45198900648350101344871774467, 3.92730428905759120918986563472, 5.02282969313034101720477689346, 6.46475745366553672339604706660, 6.76713260305245154917799835909, 8.154257407488827919731846671295, 8.476742609318584629657667383223, 9.918459466787023661418179798093, 10.44341983694093757607692897197

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