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Properties

Label 2-507-39.11-c1-0-11
Degree 22
Conductor 507507
Sign 0.916+0.399i0.916 + 0.399i
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  + (−1.45 − 0.389i)2-s + (−1.60 − 0.650i)3-s + (0.232 + 0.133i)4-s + (1.06 − 1.06i)5-s + (2.08 + 1.57i)6-s + (−0.366 − 1.36i)7-s + (1.84 + 1.84i)8-s + (2.15 + 2.08i)9-s + (−1.96 + 1.13i)10-s + (−1.06 + 3.97i)11-s + (−0.285 − 0.366i)12-s + 2.12i·14-s + (−2.40 + 1.01i)15-s + (−2.23 − 3.86i)16-s + (−2.51 + 4.36i)17-s + (−2.31 − 3.87i)18-s + ⋯
L(s)  = 1  + (−1.02 − 0.275i)2-s + (−0.926 − 0.375i)3-s + (0.116 + 0.0669i)4-s + (0.476 − 0.476i)5-s + (0.849 + 0.641i)6-s + (−0.138 − 0.516i)7-s + (0.652 + 0.652i)8-s + (0.717 + 0.696i)9-s + (−0.621 + 0.358i)10-s + (−0.321 + 1.19i)11-s + (−0.0823 − 0.105i)12-s + 0.569i·14-s + (−0.620 + 0.262i)15-s + (−0.558 − 0.966i)16-s + (−0.611 + 1.05i)17-s + (−0.546 − 0.913i)18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.916+0.399i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((0.916+0.399i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.916+0.399i0.916 + 0.399i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(89,⋅)\chi_{507} (89, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), 0.916+0.399i)(2,\ 507,\ (\ :1/2),\ 0.916 + 0.399i)

Particular Values

L(1)L(1) ≈\approx 0.561303−0.116953i0.561303 - 0.116953i
L(12)L(\frac12) ≈\approx 0.561303−0.116953i0.561303 - 0.116953i
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+(1.60+0.650i)T 1 + (1.60 + 0.650i)T
13 1 1
good2 1+(1.45+0.389i)T+(1.73+i)T2 1 + (1.45 + 0.389i)T + (1.73 + i)T^{2}
5 1+(−1.06+1.06i)T−5iT2 1 + (-1.06 + 1.06i)T - 5iT^{2}
7 1+(0.366+1.36i)T+(−6.06+3.5i)T2 1 + (0.366 + 1.36i)T + (-6.06 + 3.5i)T^{2}
11 1+(1.06−3.97i)T+(−9.52−5.5i)T2 1 + (1.06 - 3.97i)T + (-9.52 - 5.5i)T^{2}
17 1+(2.51−4.36i)T+(−8.5−14.7i)T2 1 + (2.51 - 4.36i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.73+i)T+(16.4−9.5i)T2 1 + (-3.73 + i)T + (16.4 - 9.5i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1+(−6.20+3.58i)T+(14.5−25.1i)T2 1 + (-6.20 + 3.58i)T + (14.5 - 25.1i)T^{2}
31 1+(−2.46−2.46i)T+31iT2 1 + (-2.46 - 2.46i)T + 31iT^{2}
37 1+(−5.23−1.40i)T+(32.0+18.5i)T2 1 + (-5.23 - 1.40i)T + (32.0 + 18.5i)T^{2}
41 1+(−5.42−1.45i)T+(35.5+20.5i)T2 1 + (-5.42 - 1.45i)T + (35.5 + 20.5i)T^{2}
43 1+(−1.90−1.09i)T+(21.5+37.2i)T2 1 + (-1.90 - 1.09i)T + (21.5 + 37.2i)T^{2}
47 1+(4.25+4.25i)T+47iT2 1 + (4.25 + 4.25i)T + 47iT^{2}
53 1−0.779iT−53T2 1 - 0.779iT - 53T^{2}
59 1+(2.90−0.779i)T+(51.0−29.5i)T2 1 + (2.90 - 0.779i)T + (51.0 - 29.5i)T^{2}
61 1+(−3.5+6.06i)T+(−30.5−52.8i)T2 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.53+5.73i)T+(−58.0−33.5i)T2 1 + (-1.53 + 5.73i)T + (-58.0 - 33.5i)T^{2}
71 1+(0.779+2.90i)T+(−61.4+35.5i)T2 1 + (0.779 + 2.90i)T + (-61.4 + 35.5i)T^{2}
73 1+(−0.901+0.901i)T−73iT2 1 + (-0.901 + 0.901i)T - 73iT^{2}
79 1−2T+79T2 1 - 2T + 79T^{2}
83 1+(−2.90+2.90i)T−83iT2 1 + (-2.90 + 2.90i)T - 83iT^{2}
89 1+(2.41−9.01i)T+(−77.0−44.5i)T2 1 + (2.41 - 9.01i)T + (-77.0 - 44.5i)T^{2}
97 1+(1.63−0.437i)T+(84.0−48.5i)T2 1 + (1.63 - 0.437i)T + (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

−10.62083182233852951364151609303, −10.01275702852515209963199480804, −9.349035895468912097547800320848, −8.155841564941618601880883036524, −7.37953200151122792446333508474, −6.36801986372368586872849914739, −5.15686631381056376594741499013, −4.43171567307234226472407754072, −2.09096545732883971089114848175, −0.983634087168849504202564718852, 0.75728726497467023247403713233, 2.88517853406774644525367362005, 4.40809699269477077122043360799, 5.60406148843847821954800400757, 6.40890587083136616057635038400, 7.32929989157006025081557375185, 8.462572128343446032688050737120, 9.323318220253235222340579629436, 9.973059537417069492220687684941, 10.79223158374223657960057829009

Graph of the ZZ-function along the critical line