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Properties

Label 2-507-39.20-c1-0-5
Degree 22
Conductor 507507
Sign −0.533+0.846i-0.533 + 0.846i
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  + (0.389 + 1.45i)2-s + (−0.866 + 1.5i)3-s + (−0.232 + 0.133i)4-s + (−2.90 + 2.90i)5-s + (−2.51 − 0.675i)6-s + (1.84 + 1.84i)8-s + (−1.5 − 2.59i)9-s + (−5.36 − 3.09i)10-s + (−1.06 + 0.285i)11-s − 0.464i·12-s + (−1.84 − 6.88i)15-s + (−2.23 + 3.86i)16-s + (3.19 − 3.19i)18-s + (0.285 − 1.06i)20-s + (−0.830 − 1.43i)22-s + ⋯
L(s)  = 1  + (0.275 + 1.02i)2-s + (−0.499 + 0.866i)3-s + (−0.116 + 0.0669i)4-s + (−1.30 + 1.30i)5-s + (−1.02 − 0.275i)6-s + (0.652 + 0.652i)8-s + (−0.5 − 0.866i)9-s + (−1.69 − 0.979i)10-s + (−0.321 + 0.0860i)11-s − 0.133i·12-s + (−0.476 − 1.77i)15-s + (−0.558 + 0.966i)16-s + (0.752 − 0.752i)18-s + (0.0638 − 0.238i)20-s + (−0.176 − 0.306i)22-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.533+0.846i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.533+0.846i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.533+0.846i-0.533 + 0.846i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(488,⋅)\chi_{507} (488, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.533+0.846i)(2,\ 507,\ (\ :1/2),\ -0.533 + 0.846i)

Particular Values

L(1)L(1) ≈\approx 0.407783−0.738888i0.407783 - 0.738888i
L(12)L(\frac12) ≈\approx 0.407783−0.738888i0.407783 - 0.738888i
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+(0.866−1.5i)T 1 + (0.866 - 1.5i)T
13 1 1
good2 1+(−0.389−1.45i)T+(−1.73+i)T2 1 + (-0.389 - 1.45i)T + (-1.73 + i)T^{2}
5 1+(2.90−2.90i)T−5iT2 1 + (2.90 - 2.90i)T - 5iT^{2}
7 1+(6.06+3.5i)T2 1 + (6.06 + 3.5i)T^{2}
11 1+(1.06−0.285i)T+(9.52−5.5i)T2 1 + (1.06 - 0.285i)T + (9.52 - 5.5i)T^{2}
17 1+(−8.5+14.7i)T2 1 + (-8.5 + 14.7i)T^{2}
19 1+(−16.4−9.5i)T2 1 + (-16.4 - 9.5i)T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+(14.5+25.1i)T2 1 + (14.5 + 25.1i)T^{2}
31 1+31iT2 1 + 31iT^{2}
37 1+(−32.0+18.5i)T2 1 + (-32.0 + 18.5i)T^{2}
41 1+(−2.62−9.79i)T+(−35.5+20.5i)T2 1 + (-2.62 - 9.79i)T + (-35.5 + 20.5i)T^{2}
43 1+(3.46−2i)T+(21.5−37.2i)T2 1 + (3.46 - 2i)T + (21.5 - 37.2i)T^{2}
47 1+(6.59+6.59i)T+47iT2 1 + (6.59 + 6.59i)T + 47iT^{2}
53 1−53T2 1 - 53T^{2}
59 1+(3.97−14.8i)T+(−51.0−29.5i)T2 1 + (3.97 - 14.8i)T + (-51.0 - 29.5i)T^{2}
61 1+(−6.92−12i)T+(−30.5+52.8i)T2 1 + (-6.92 - 12i)T + (-30.5 + 52.8i)T^{2}
67 1+(58.0−33.5i)T2 1 + (58.0 - 33.5i)T^{2}
71 1+(4.75+1.27i)T+(61.4+35.5i)T2 1 + (4.75 + 1.27i)T + (61.4 + 35.5i)T^{2}
73 1−73iT2 1 - 73iT^{2}
79 1−10.3T+79T2 1 - 10.3T + 79T^{2}
83 1+(−9.29+9.29i)T−83iT2 1 + (-9.29 + 9.29i)T - 83iT^{2}
89 1+(17.7−4.75i)T+(77.0−44.5i)T2 1 + (17.7 - 4.75i)T + (77.0 - 44.5i)T^{2}
97 1+(−84.0−48.5i)T2 1 + (-84.0 - 48.5i)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

−11.38063440062360944667527971963, −10.69920591405193129139486670476, −9.990077169939600508914228775188, −8.495199275477907060240920566752, −7.66644114918838644346555336856, −6.82976325543520716425656325428, −6.12252217093397613492932112369, −4.95338289364915345740152474156, −4.03409763126126872004764144372, −2.91522817695959817203080778591, 0.48634814710896855157750741463, 1.76113247808402010973505500028, 3.30848896402607797933742085126, 4.44691056243219158394704734435, 5.28906367053479571257649717561, 6.78304921080310488546844998512, 7.75240836376866608702198848389, 8.313021387241324470700621438967, 9.541156529431912016578264630718, 10.91173815584280209781114506104

Graph of the ZZ-function along the critical line