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Properties

Label 2-507-13.10-c1-0-13
Degree 22
Conductor 507507
Sign 0.964−0.265i0.964 - 0.265i
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  + (−1.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (−1.5 − 0.866i)6-s + (−3 − 1.73i)7-s − 1.73i·8-s + (−0.499 + 0.866i)9-s + (3 − 1.73i)11-s + 12-s + 6·14-s + (2.49 + 4.33i)16-s + (3 − 5.19i)17-s − 1.73i·18-s + (3 + 1.73i)19-s − 3.46i·21-s + (−3 + 5.19i)22-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s + (0.288 + 0.499i)3-s + (0.250 − 0.433i)4-s + (−0.612 − 0.353i)6-s + (−1.13 − 0.654i)7-s − 0.612i·8-s + (−0.166 + 0.288i)9-s + (0.904 − 0.522i)11-s + 0.288·12-s + 1.60·14-s + (0.624 + 1.08i)16-s + (0.727 − 1.26i)17-s − 0.408i·18-s + (0.688 + 0.397i)19-s − 0.755i·21-s + (−0.639 + 1.10i)22-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.964−0.265i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((0.964−0.265i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.964−0.265i0.964 - 0.265i
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), 0.964−0.265i)(2,\ 507,\ (\ :1/2),\ 0.964 - 0.265i)

Particular Values

L(1)L(1) ≈\approx 0.757699+0.102225i0.757699 + 0.102225i
L(12)L(\frac12) ≈\approx 0.757699+0.102225i0.757699 + 0.102225i
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+(1.5−0.866i)T+(1−1.73i)T2 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2}
5 1−5T2 1 - 5T^{2}
7 1+(3+1.73i)T+(3.5+6.06i)T2 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2}
11 1+(−3+1.73i)T+(5.5−9.52i)T2 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2}
17 1+(−3+5.19i)T+(−8.5−14.7i)T2 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3−1.73i)T+(9.5+16.4i)T2 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1+(3+5.19i)T+(−14.5+25.1i)T2 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2}
31 1−3.46iT−31T2 1 - 3.46iT - 31T^{2}
37 1+(6−3.46i)T+(18.5−32.0i)T2 1 + (6 - 3.46i)T + (18.5 - 32.0i)T^{2}
41 1+(−6+3.46i)T+(20.5−35.5i)T2 1 + (-6 + 3.46i)T + (20.5 - 35.5i)T^{2}
43 1+(−2+3.46i)T+(−21.5−37.2i)T2 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2}
47 1+3.46iT−47T2 1 + 3.46iT - 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1+(−9−5.19i)T+(29.5+51.0i)T2 1 + (-9 - 5.19i)T + (29.5 + 51.0i)T^{2}
61 1+(−1+1.73i)T+(−30.5−52.8i)T2 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2}
67 1+(−9+5.19i)T+(33.5−58.0i)T2 1 + (-9 + 5.19i)T + (33.5 - 58.0i)T^{2}
71 1+(−3−1.73i)T+(35.5+61.4i)T2 1 + (-3 - 1.73i)T + (35.5 + 61.4i)T^{2}
73 1−73T2 1 - 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1−3.46iT−83T2 1 - 3.46iT - 83T^{2}
89 1+(−6+3.46i)T+(44.5−77.0i)T2 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2}
97 1+(12+6.92i)T+(48.5+84.0i)T2 1 + (12 + 6.92i)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

−10.45737014804363880044918247950, −9.784279867347200058000851551102, −9.238730487837506726398540269362, −8.424945101879720860170709452951, −7.29663959649544371009586103992, −6.77056571878159983454859394878, −5.56565213639817030126170087840, −3.95810825017970346184274956770, −3.19004994129091839680150281350, −0.75791641788707269968501172580, 1.21799310853268573511548543471, 2.50052401206649616738105829340, 3.62528122291624945926772322282, 5.42793168713795955413936074046, 6.45756177055693616703738269197, 7.43514340593631472008992573431, 8.521584955690817098320288158146, 9.222001013497993981012487905346, 9.745189651565705237300685453805, 10.70823101381231867160946177586

Graph of the ZZ-function along the critical line