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Properties

Label 2-592-37.34-c1-0-5
Degree $2$
Conductor $592$
Sign $0.512 - 0.858i$
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  + (−0.149 + 0.125i)3-s + (2.90 + 1.05i)5-s + (0.434 + 0.158i)7-s + (−0.514 + 2.91i)9-s + (−1.13 + 1.96i)11-s + (0.321 + 1.82i)13-s + (−0.567 + 0.206i)15-s + (0.342 − 1.94i)17-s + (−1.00 + 0.840i)19-s + (−0.0848 + 0.0308i)21-s + (−0.276 − 0.478i)23-s + (3.49 + 2.93i)25-s + (−0.582 − 1.00i)27-s + (0.286 − 0.496i)29-s − 1.10·31-s + ⋯
L(s)  = 1  + (−0.0864 + 0.0725i)3-s + (1.29 + 0.473i)5-s + (0.164 + 0.0597i)7-s + (−0.171 + 0.972i)9-s + (−0.342 + 0.593i)11-s + (0.0892 + 0.506i)13-s + (−0.146 + 0.0533i)15-s + (0.0830 − 0.470i)17-s + (−0.229 + 0.192i)19-s + (−0.0185 + 0.00673i)21-s + (−0.0575 − 0.0997i)23-s + (0.699 + 0.586i)25-s + (−0.112 − 0.194i)27-s + (0.0532 − 0.0922i)29-s − 0.198·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.512 - 0.858i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48271 + 0.841676i\)
\(L(\frac12)\) \(\approx\) \(1.48271 + 0.841676i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.95 + 3.52i)T \)
good3 \( 1 + (0.149 - 0.125i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (-2.90 - 1.05i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.434 - 0.158i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (1.13 - 1.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.321 - 1.82i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.342 + 1.94i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (1.00 - 0.840i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (0.276 + 0.478i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.286 + 0.496i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.10T + 31T^{2} \)
41 \( 1 + (-0.616 - 3.49i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 - 5.87T + 43T^{2} \)
47 \( 1 + (-5.57 - 9.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.14 - 2.23i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-11.5 + 4.22i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.07 + 6.07i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-12.9 - 4.69i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (0.289 - 0.243i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 2.77T + 73T^{2} \)
79 \( 1 + (11.5 + 4.21i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-2.17 + 12.3i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-8.54 + 3.11i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (8.17 + 14.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.76550368289984075457144172590, −9.934526404872984546457703967027, −9.340792723309739003430828152851, −8.146298221528434440483206064059, −7.21870331452715883888775443905, −6.20117615610610330455009569896, −5.38709415114114533633007130554, −4.42460233959924073745097379161, −2.68787678110805073760929701197, −1.87710647056624981133999695751, 1.03092173998373313570215561420, 2.48713848166033538338511448394, 3.81780978373814895454751845671, 5.26438949530023554687082348312, 5.88439211172962064877431789357, 6.71664284351635214954773767076, 8.061260812477573984891639778430, 8.900681430200718351863658051668, 9.630303300200111364955080510866, 10.44287909485693195408832038974

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