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Properties

Label 2-592-37.12-c1-0-5
Degree 22
Conductor 592592
Sign 0.998−0.0521i0.998 - 0.0521i
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  + (−2.29 − 1.92i)3-s + (1.16 − 0.423i)5-s + (−1.37 + 0.501i)7-s + (1.03 + 5.84i)9-s + (3.02 + 5.23i)11-s + (−0.652 + 3.70i)13-s + (−3.48 − 1.26i)15-s + (−0.936 − 5.30i)17-s + (2.77 + 2.32i)19-s + (4.12 + 1.50i)21-s + (2.92 − 5.07i)23-s + (−2.65 + 2.22i)25-s + (4.39 − 7.61i)27-s + (2.60 + 4.50i)29-s + 2.33·31-s + ⋯
L(s)  = 1  + (−1.32 − 1.10i)3-s + (0.520 − 0.189i)5-s + (−0.521 + 0.189i)7-s + (0.343 + 1.94i)9-s + (0.912 + 1.57i)11-s + (−0.180 + 1.02i)13-s + (−0.899 − 0.327i)15-s + (−0.227 − 1.28i)17-s + (0.635 + 0.533i)19-s + (0.899 + 0.327i)21-s + (0.610 − 1.05i)23-s + (−0.530 + 0.445i)25-s + (0.845 − 1.46i)27-s + (0.483 + 0.836i)29-s + 0.419·31-s + ⋯

Functional equation

Λ(s)=(592s/2 ΓC(s) L(s)=((0.998−0.0521i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0521i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(592s/2 ΓC(s+1/2) L(s)=((0.998−0.0521i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 592592    =    24⋅372^{4} \cdot 37
Sign: 0.998−0.0521i0.998 - 0.0521i
Analytic conductor: 4.727144.72714
Root analytic conductor: 2.174192.17419
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ592(49,⋅)\chi_{592} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 592, ( :1/2), 0.998−0.0521i)(2,\ 592,\ (\ :1/2),\ 0.998 - 0.0521i)

Particular Values

L(1)L(1) ≈\approx 0.937750+0.0244601i0.937750 + 0.0244601i
L(12)L(\frac12) ≈\approx 0.937750+0.0244601i0.937750 + 0.0244601i
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+(−5.43+2.73i)T 1 + (-5.43 + 2.73i)T
good3 1+(2.29+1.92i)T+(0.520+2.95i)T2 1 + (2.29 + 1.92i)T + (0.520 + 2.95i)T^{2}
5 1+(−1.16+0.423i)T+(3.83−3.21i)T2 1 + (-1.16 + 0.423i)T + (3.83 - 3.21i)T^{2}
7 1+(1.37−0.501i)T+(5.36−4.49i)T2 1 + (1.37 - 0.501i)T + (5.36 - 4.49i)T^{2}
11 1+(−3.02−5.23i)T+(−5.5+9.52i)T2 1 + (-3.02 - 5.23i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.652−3.70i)T+(−12.2−4.44i)T2 1 + (0.652 - 3.70i)T + (-12.2 - 4.44i)T^{2}
17 1+(0.936+5.30i)T+(−15.9+5.81i)T2 1 + (0.936 + 5.30i)T + (-15.9 + 5.81i)T^{2}
19 1+(−2.77−2.32i)T+(3.29+18.7i)T2 1 + (-2.77 - 2.32i)T + (3.29 + 18.7i)T^{2}
23 1+(−2.92+5.07i)T+(−11.5−19.9i)T2 1 + (-2.92 + 5.07i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.60−4.50i)T+(−14.5+25.1i)T2 1 + (-2.60 - 4.50i)T + (-14.5 + 25.1i)T^{2}
31 1−2.33T+31T2 1 - 2.33T + 31T^{2}
41 1+(−0.173+0.983i)T+(−38.5−14.0i)T2 1 + (-0.173 + 0.983i)T + (-38.5 - 14.0i)T^{2}
43 1−5.45T+43T2 1 - 5.45T + 43T^{2}
47 1+(3.81−6.60i)T+(−23.5−40.7i)T2 1 + (3.81 - 6.60i)T + (-23.5 - 40.7i)T^{2}
53 1+(−8.54−3.11i)T+(40.6+34.0i)T2 1 + (-8.54 - 3.11i)T + (40.6 + 34.0i)T^{2}
59 1+(7.89+2.87i)T+(45.1+37.9i)T2 1 + (7.89 + 2.87i)T + (45.1 + 37.9i)T^{2}
61 1+(−0.810+4.59i)T+(−57.3−20.8i)T2 1 + (-0.810 + 4.59i)T + (-57.3 - 20.8i)T^{2}
67 1+(0.757−0.275i)T+(51.3−43.0i)T2 1 + (0.757 - 0.275i)T + (51.3 - 43.0i)T^{2}
71 1+(−10.6−8.93i)T+(12.3+69.9i)T2 1 + (-10.6 - 8.93i)T + (12.3 + 69.9i)T^{2}
73 1−7.11T+73T2 1 - 7.11T + 73T^{2}
79 1+(−6.61+2.40i)T+(60.5−50.7i)T2 1 + (-6.61 + 2.40i)T + (60.5 - 50.7i)T^{2}
83 1+(−0.942−5.34i)T+(−77.9+28.3i)T2 1 + (-0.942 - 5.34i)T + (-77.9 + 28.3i)T^{2}
89 1+(5.21+1.89i)T+(68.1+57.2i)T2 1 + (5.21 + 1.89i)T + (68.1 + 57.2i)T^{2}
97 1+(5.35−9.28i)T+(−48.5−84.0i)T2 1 + (5.35 - 9.28i)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

−10.93353690446046518461843682823, −9.677747425810671126369641901567, −9.272366426728922542824655465052, −7.61216681330711702648745984948, −6.83278053304944233501430132060, −6.42382327642804474640034119578, −5.27655715075896003541215605383, −4.45340553173839751552370635283, −2.34696036094569442975761936886, −1.21973427834495324668188788907, 0.74293730252705299474327550520, 3.20805818348863545423541603435, 4.06394832029989119243962835385, 5.35450682896020785427402767180, 6.00806931023344353080360595884, 6.56732965896653573965797679180, 8.164020246849354251016942201633, 9.308289036713100192202328238029, 9.957426636600081931997909105542, 10.68045797311217385263049085132

Graph of the ZZ-function along the critical line