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Properties

Label 2-592-37.33-c1-0-9
Degree 22
Conductor 592592
Sign 0.886−0.462i0.886 - 0.462i
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  + (0.260 + 1.47i)3-s + (−1.75 + 1.47i)5-s + (3.54 − 2.97i)7-s + (0.709 − 0.258i)9-s + (2.89 − 5.02i)11-s + (1.41 + 0.513i)13-s + (−2.62 − 2.20i)15-s + (−0.343 + 0.125i)17-s + (0.769 + 4.36i)19-s + (5.30 + 4.45i)21-s + (−1.15 − 1.99i)23-s + (0.0400 − 0.227i)25-s + (2.81 + 4.87i)27-s + (−1.46 + 2.53i)29-s − 0.428·31-s + ⋯
L(s)  = 1  + (0.150 + 0.851i)3-s + (−0.783 + 0.657i)5-s + (1.33 − 1.12i)7-s + (0.236 − 0.0860i)9-s + (0.874 − 1.51i)11-s + (0.391 + 0.142i)13-s + (−0.677 − 0.568i)15-s + (−0.0833 + 0.0303i)17-s + (0.176 + 1.00i)19-s + (1.15 + 0.971i)21-s + (−0.240 − 0.416i)23-s + (0.00800 − 0.0454i)25-s + (0.541 + 0.937i)27-s + (−0.271 + 0.470i)29-s − 0.0769·31-s + ⋯

Functional equation

Λ(s)=(592s/2 ΓC(s) L(s)=((0.886−0.462i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(592s/2 ΓC(s+1/2) L(s)=((0.886−0.462i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 592592    =    24⋅372^{4} \cdot 37
Sign: 0.886−0.462i0.886 - 0.462i
Analytic conductor: 4.727144.72714
Root analytic conductor: 2.174192.17419
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ592(33,⋅)\chi_{592} (33, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 592, ( :1/2), 0.886−0.462i)(2,\ 592,\ (\ :1/2),\ 0.886 - 0.462i)

Particular Values

L(1)L(1) ≈\approx 1.65033+0.404710i1.65033 + 0.404710i
L(12)L(\frac12) ≈\approx 1.65033+0.404710i1.65033 + 0.404710i
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+(3.07−5.24i)T 1 + (3.07 - 5.24i)T
good3 1+(−0.260−1.47i)T+(−2.81+1.02i)T2 1 + (-0.260 - 1.47i)T + (-2.81 + 1.02i)T^{2}
5 1+(1.75−1.47i)T+(0.868−4.92i)T2 1 + (1.75 - 1.47i)T + (0.868 - 4.92i)T^{2}
7 1+(−3.54+2.97i)T+(1.21−6.89i)T2 1 + (-3.54 + 2.97i)T + (1.21 - 6.89i)T^{2}
11 1+(−2.89+5.02i)T+(−5.5−9.52i)T2 1 + (-2.89 + 5.02i)T + (-5.5 - 9.52i)T^{2}
13 1+(−1.41−0.513i)T+(9.95+8.35i)T2 1 + (-1.41 - 0.513i)T + (9.95 + 8.35i)T^{2}
17 1+(0.343−0.125i)T+(13.0−10.9i)T2 1 + (0.343 - 0.125i)T + (13.0 - 10.9i)T^{2}
19 1+(−0.769−4.36i)T+(−17.8+6.49i)T2 1 + (-0.769 - 4.36i)T + (-17.8 + 6.49i)T^{2}
23 1+(1.15+1.99i)T+(−11.5+19.9i)T2 1 + (1.15 + 1.99i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.46−2.53i)T+(−14.5−25.1i)T2 1 + (1.46 - 2.53i)T + (-14.5 - 25.1i)T^{2}
31 1+0.428T+31T2 1 + 0.428T + 31T^{2}
41 1+(−0.112−0.0408i)T+(31.4+26.3i)T2 1 + (-0.112 - 0.0408i)T + (31.4 + 26.3i)T^{2}
43 1−12.9T+43T2 1 - 12.9T + 43T^{2}
47 1+(3.30+5.72i)T+(−23.5+40.7i)T2 1 + (3.30 + 5.72i)T + (-23.5 + 40.7i)T^{2}
53 1+(7.42+6.22i)T+(9.20+52.1i)T2 1 + (7.42 + 6.22i)T + (9.20 + 52.1i)T^{2}
59 1+(6.22+5.22i)T+(10.2+58.1i)T2 1 + (6.22 + 5.22i)T + (10.2 + 58.1i)T^{2}
61 1+(−2.40−0.874i)T+(46.7+39.2i)T2 1 + (-2.40 - 0.874i)T + (46.7 + 39.2i)T^{2}
67 1+(10.8−9.07i)T+(11.6−65.9i)T2 1 + (10.8 - 9.07i)T + (11.6 - 65.9i)T^{2}
71 1+(1.40+7.98i)T+(−66.7+24.2i)T2 1 + (1.40 + 7.98i)T + (-66.7 + 24.2i)T^{2}
73 1−13.2T+73T2 1 - 13.2T + 73T^{2}
79 1+(8.48−7.12i)T+(13.7−77.7i)T2 1 + (8.48 - 7.12i)T + (13.7 - 77.7i)T^{2}
83 1+(−6.92+2.52i)T+(63.5−53.3i)T2 1 + (-6.92 + 2.52i)T + (63.5 - 53.3i)T^{2}
89 1+(−10.7−9.02i)T+(15.4+87.6i)T2 1 + (-10.7 - 9.02i)T + (15.4 + 87.6i)T^{2}
97 1+(2.83+4.90i)T+(−48.5+84.0i)T2 1 + (2.83 + 4.90i)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

−10.96528262040743638331547300273, −10.10526800520036899401646909114, −8.918650728172382945064342775346, −8.123743462507701720131701681577, −7.33263322375550131581649491649, −6.28734287629448944066204213068, −4.90716536626528527290024752003, −3.83909386075898937713263385289, −3.56163704490602141431086063574, −1.27632098674875133389341141089, 1.37575179012117683143608810099, 2.29470563041123978010234298254, 4.26970399755972335178668226877, 4.84189077166854722393917075636, 6.13142598495861545794251354932, 7.41465163364718690344700095397, 7.77624822038151839828244989448, 8.799809107682927273142670497576, 9.401218492750010302869322535414, 10.89461186624953965378094864298

Graph of the ZZ-function along the critical line