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Properties

Label 2-507-13.10-c1-0-20
Degree 22
Conductor 507507
Sign 0.455+0.890i0.455 + 0.890i
Analytic cond. 4.048414.04841
Root an. cond. 2.012062.01206
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  + (−2.04 + 1.17i)2-s + (0.5 + 0.866i)3-s + (1.77 − 3.07i)4-s − 3.69i·5-s + (−2.04 − 1.17i)6-s + (0.694 + 0.400i)7-s + 3.66i·8-s + (−0.499 + 0.866i)9-s + (4.35 + 7.53i)10-s + (−2.46 + 1.42i)11-s + 3.55·12-s − 1.89·14-s + (3.19 − 1.84i)15-s + (−0.763 − 1.32i)16-s + (1.46 − 2.54i)17-s − 2.35i·18-s + ⋯
L(s)  = 1  + (−1.44 + 0.833i)2-s + (0.288 + 0.499i)3-s + (0.888 − 1.53i)4-s − 1.65i·5-s + (−0.833 − 0.481i)6-s + (0.262 + 0.151i)7-s + 1.29i·8-s + (−0.166 + 0.288i)9-s + (1.37 + 2.38i)10-s + (−0.744 + 0.429i)11-s + 1.02·12-s − 0.505·14-s + (0.825 − 0.476i)15-s + (−0.190 − 0.330i)16-s + (0.356 − 0.617i)17-s − 0.555i·18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.455+0.890i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((0.455+0.890i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.455+0.890i0.455 + 0.890i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(361,⋅)\chi_{507} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), 0.455+0.890i)(2,\ 507,\ (\ :1/2),\ 0.455 + 0.890i)

Particular Values

L(1)L(1) ≈\approx 0.435108−0.266019i0.435108 - 0.266019i
L(12)L(\frac12) ≈\approx 0.435108−0.266019i0.435108 - 0.266019i
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)
bad3 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
13 1 1
good2 1+(2.04−1.17i)T+(1−1.73i)T2 1 + (2.04 - 1.17i)T + (1 - 1.73i)T^{2}
5 1+3.69iT−5T2 1 + 3.69iT - 5T^{2}
7 1+(−0.694−0.400i)T+(3.5+6.06i)T2 1 + (-0.694 - 0.400i)T + (3.5 + 6.06i)T^{2}
11 1+(2.46−1.42i)T+(5.5−9.52i)T2 1 + (2.46 - 1.42i)T + (5.5 - 9.52i)T^{2}
17 1+(−1.46+2.54i)T+(−8.5−14.7i)T2 1 + (-1.46 + 2.54i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.11+1.22i)T+(9.5+16.4i)T2 1 + (2.11 + 1.22i)T + (9.5 + 16.4i)T^{2}
23 1+(3.89+6.74i)T+(−11.5+19.9i)T2 1 + (3.89 + 6.74i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.92+3.33i)T+(−14.5+25.1i)T2 1 + (1.92 + 3.33i)T + (-14.5 + 25.1i)T^{2}
31 1+2.34iT−31T2 1 + 2.34iT - 31T^{2}
37 1+(−6.44+3.72i)T+(18.5−32.0i)T2 1 + (-6.44 + 3.72i)T + (18.5 - 32.0i)T^{2}
41 1+(−0.736+0.425i)T+(20.5−35.5i)T2 1 + (-0.736 + 0.425i)T + (20.5 - 35.5i)T^{2}
43 1+(0.807−1.39i)T+(−21.5−37.2i)T2 1 + (0.807 - 1.39i)T + (-21.5 - 37.2i)T^{2}
47 1−2.44iT−47T2 1 - 2.44iT - 47T^{2}
53 1+9.96T+53T2 1 + 9.96T + 53T^{2}
59 1+(−4.66−2.69i)T+(29.5+51.0i)T2 1 + (-4.66 - 2.69i)T + (29.5 + 51.0i)T^{2}
61 1+(−6.62+11.4i)T+(−30.5−52.8i)T2 1 + (-6.62 + 11.4i)T + (-30.5 - 52.8i)T^{2}
67 1+(12.4−7.19i)T+(33.5−58.0i)T2 1 + (12.4 - 7.19i)T + (33.5 - 58.0i)T^{2}
71 1+(−7.03−4.06i)T+(35.5+61.4i)T2 1 + (-7.03 - 4.06i)T + (35.5 + 61.4i)T^{2}
73 1+11.8iT−73T2 1 + 11.8iT - 73T^{2}
79 1−5.40T+79T2 1 - 5.40T + 79T^{2}
83 1+7.04iT−83T2 1 + 7.04iT - 83T^{2}
89 1+(−0.980+0.565i)T+(44.5−77.0i)T2 1 + (-0.980 + 0.565i)T + (44.5 - 77.0i)T^{2}
97 1+(5.14+2.97i)T+(48.5+84.0i)T2 1 + (5.14 + 2.97i)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

−10.24094942665387753025999034451, −9.572714573931590088901959036782, −8.915225298531381763939975227426, −8.154225159643601055499899246553, −7.69304719048801530127815822352, −6.23184189177478403338543008267, −5.19758395826911196279200305482, −4.34838774373899056277673870030, −2.10227553599713640332473178323, −0.45991190979684038426009964869, 1.67828296770557678097827034928, 2.75828251641600142317344336650, 3.56560902640320203036525017865, 5.83934673171575046384065996799, 6.94970474369351385533375332311, 7.77804505485784488894905978475, 8.242648406883936580186942292406, 9.523729804909872185004053238413, 10.24979141157167443584342030334, 10.92675639530558140026838741364

Graph of the ZZ-function along the critical line