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Properties

Label 2-592-37.34-c1-0-5
Degree 22
Conductor 592592
Sign 0.512−0.858i0.512 - 0.858i
Analytic cond. 4.727144.72714
Root an. cond. 2.174192.17419
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(592s/2 ΓC(s) L(s)=((0.512−0.858i) Λ‾(2−s)\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}
Λ(s)=(592s/2 ΓC(s+1/2) L(s)=((0.512−0.858i) Λ‾(1−s)\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: 22
Conductor: 592592    =    24⋅372^{4} \cdot 37
Sign: 0.512−0.858i0.512 - 0.858i
Analytic conductor: 4.727144.72714
Root analytic conductor: 2.174192.17419
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ592(145,⋅)\chi_{592} (145, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 592, ( :1/2), 0.512−0.858i)(2,\ 592,\ (\ :1/2),\ 0.512 - 0.858i)

Particular Values

L(1)L(1) ≈\approx 1.48271+0.841676i1.48271 + 0.841676i
L(12)L(\frac12) ≈\approx 1.48271+0.841676i1.48271 + 0.841676i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
37 1+(−4.95+3.52i)T 1 + (-4.95 + 3.52i)T
good3 1+(0.149−0.125i)T+(0.520−2.95i)T2 1 + (0.149 - 0.125i)T + (0.520 - 2.95i)T^{2}
5 1+(−2.90−1.05i)T+(3.83+3.21i)T2 1 + (-2.90 - 1.05i)T + (3.83 + 3.21i)T^{2}
7 1+(−0.434−0.158i)T+(5.36+4.49i)T2 1 + (-0.434 - 0.158i)T + (5.36 + 4.49i)T^{2}
11 1+(1.13−1.96i)T+(−5.5−9.52i)T2 1 + (1.13 - 1.96i)T + (-5.5 - 9.52i)T^{2}
13 1+(−0.321−1.82i)T+(−12.2+4.44i)T2 1 + (-0.321 - 1.82i)T + (-12.2 + 4.44i)T^{2}
17 1+(−0.342+1.94i)T+(−15.9−5.81i)T2 1 + (-0.342 + 1.94i)T + (-15.9 - 5.81i)T^{2}
19 1+(1.00−0.840i)T+(3.29−18.7i)T2 1 + (1.00 - 0.840i)T + (3.29 - 18.7i)T^{2}
23 1+(0.276+0.478i)T+(−11.5+19.9i)T2 1 + (0.276 + 0.478i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.286+0.496i)T+(−14.5−25.1i)T2 1 + (-0.286 + 0.496i)T + (-14.5 - 25.1i)T^{2}
31 1+1.10T+31T2 1 + 1.10T + 31T^{2}
41 1+(−0.616−3.49i)T+(−38.5+14.0i)T2 1 + (-0.616 - 3.49i)T + (-38.5 + 14.0i)T^{2}
43 1−5.87T+43T2 1 - 5.87T + 43T^{2}
47 1+(−5.57−9.66i)T+(−23.5+40.7i)T2 1 + (-5.57 - 9.66i)T + (-23.5 + 40.7i)T^{2}
53 1+(6.14−2.23i)T+(40.6−34.0i)T2 1 + (6.14 - 2.23i)T + (40.6 - 34.0i)T^{2}
59 1+(−11.5+4.22i)T+(45.1−37.9i)T2 1 + (-11.5 + 4.22i)T + (45.1 - 37.9i)T^{2}
61 1+(1.07+6.07i)T+(−57.3+20.8i)T2 1 + (1.07 + 6.07i)T + (-57.3 + 20.8i)T^{2}
67 1+(−12.9−4.69i)T+(51.3+43.0i)T2 1 + (-12.9 - 4.69i)T + (51.3 + 43.0i)T^{2}
71 1+(0.289−0.243i)T+(12.3−69.9i)T2 1 + (0.289 - 0.243i)T + (12.3 - 69.9i)T^{2}
73 1−2.77T+73T2 1 - 2.77T + 73T^{2}
79 1+(11.5+4.21i)T+(60.5+50.7i)T2 1 + (11.5 + 4.21i)T + (60.5 + 50.7i)T^{2}
83 1+(−2.17+12.3i)T+(−77.9−28.3i)T2 1 + (-2.17 + 12.3i)T + (-77.9 - 28.3i)T^{2}
89 1+(−8.54+3.11i)T+(68.1−57.2i)T2 1 + (-8.54 + 3.11i)T + (68.1 - 57.2i)T^{2}
97 1+(8.17+14.1i)T+(−48.5+84.0i)T2 1 + (8.17 + 14.1i)T + (-48.5 + 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line