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Properties

Label 2-592-37.7-c1-0-8
Degree $2$
Conductor $592$
Sign $0.928 - 0.372i$
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  + (2.04 + 0.746i)3-s + (−0.326 − 1.85i)5-s + (0.711 + 4.03i)7-s + (1.34 + 1.12i)9-s + (1.67 − 2.89i)11-s + (1.48 − 1.24i)13-s + (0.711 − 4.03i)15-s + (3.40 + 2.85i)17-s + (−0.0932 − 0.0339i)19-s + (−1.55 + 8.80i)21-s + (4.76 + 8.24i)23-s + (1.37 − 0.502i)25-s + (−1.35 − 2.34i)27-s + (−0.387 + 0.670i)29-s − 10.3·31-s + ⋯
L(s)  = 1  + (1.18 + 0.430i)3-s + (−0.145 − 0.827i)5-s + (0.269 + 1.52i)7-s + (0.448 + 0.376i)9-s + (0.503 − 0.872i)11-s + (0.411 − 0.345i)13-s + (0.183 − 1.04i)15-s + (0.826 + 0.693i)17-s + (−0.0213 − 0.00778i)19-s + (−0.338 + 1.92i)21-s + (0.992 + 1.71i)23-s + (0.275 − 0.100i)25-s + (−0.260 − 0.451i)27-s + (−0.0719 + 0.124i)29-s − 1.86·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.928 - 0.372i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.928 - 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19122 + 0.423026i\)
\(L(\frac12)\) \(\approx\) \(2.19122 + 0.423026i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.50 + 4.97i)T \)
good3 \( 1 + (-2.04 - 0.746i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (0.326 + 1.85i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-0.711 - 4.03i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-1.67 + 2.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.48 + 1.24i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-3.40 - 2.85i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (0.0932 + 0.0339i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-4.76 - 8.24i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.387 - 0.670i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
41 \( 1 + (-4.36 + 3.66i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 1.11T + 43T^{2} \)
47 \( 1 + (3.55 + 6.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.142 + 0.806i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (1.40 - 7.97i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (8.19 - 6.87i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (2.28 + 12.9i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (2.79 + 1.01i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 2.55T + 73T^{2} \)
79 \( 1 + (-0.943 - 5.35i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.504 + 0.423i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-0.612 + 3.47i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.81 + 3.13i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.71634465323598659337041406014, −9.273884995297219732885630979792, −9.031664412208090689154482414416, −8.449533580426146021556913425422, −7.56298243668200263272691823383, −5.83605778854652104194918038583, −5.32653125228168175447348670039, −3.80660204956624403634676629941, −3.07564074026613861839799903016, −1.63616733013974435698629163669, 1.42356972499070718931819762798, 2.83213781054951393999395428529, 3.72993354378994177852109370073, 4.75935015159451015041373357916, 6.61014071739916718261203169718, 7.22315676718715083986108312314, 7.74432018586240747028094242807, 8.838014953240700211164001947517, 9.702993063306995141749290729361, 10.67118102700748078508052676088

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