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Properties

Label 2-507-39.8-c1-0-3
Degree 22
Conductor 507507
Sign −0.771+0.636i-0.771 + 0.636i
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  + (−0.540 + 0.540i)2-s + (0.0858 + 1.72i)3-s + 1.41i·4-s + (0.996 − 0.996i)5-s + (−0.981 − 0.888i)6-s + (−1.80 + 1.80i)7-s + (−1.84 − 1.84i)8-s + (−2.98 + 0.296i)9-s + 1.07i·10-s + (−3.35 − 3.35i)11-s + (−2.44 + 0.121i)12-s − 1.94i·14-s + (1.80 + 1.63i)15-s − 0.837·16-s − 5.80·17-s + (1.45 − 1.77i)18-s + ⋯
L(s)  = 1  + (−0.382 + 0.382i)2-s + (0.0495 + 0.998i)3-s + 0.708i·4-s + (0.445 − 0.445i)5-s + (−0.400 − 0.362i)6-s + (−0.681 + 0.681i)7-s + (−0.652 − 0.652i)8-s + (−0.995 + 0.0989i)9-s + 0.340i·10-s + (−1.01 − 1.01i)11-s + (−0.707 + 0.0350i)12-s − 0.520i·14-s + (0.467 + 0.422i)15-s − 0.209·16-s − 1.40·17-s + (0.342 − 0.417i)18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.771+0.636i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.771+0.636i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.771+0.636i-0.771 + 0.636i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(437,⋅)\chi_{507} (437, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.771+0.636i)(2,\ 507,\ (\ :1/2),\ -0.771 + 0.636i)

Particular Values

L(1)L(1) ≈\approx 0.165986−0.461815i0.165986 - 0.461815i
L(12)L(\frac12) ≈\approx 0.165986−0.461815i0.165986 - 0.461815i
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+(−0.0858−1.72i)T 1 + (-0.0858 - 1.72i)T
13 1 1
good2 1+(0.540−0.540i)T−2iT2 1 + (0.540 - 0.540i)T - 2iT^{2}
5 1+(−0.996+0.996i)T−5iT2 1 + (-0.996 + 0.996i)T - 5iT^{2}
7 1+(1.80−1.80i)T−7iT2 1 + (1.80 - 1.80i)T - 7iT^{2}
11 1+(3.35+3.35i)T+11iT2 1 + (3.35 + 3.35i)T + 11iT^{2}
17 1+5.80T+17T2 1 + 5.80T + 17T^{2}
19 1+(−2.39−2.39i)T+19iT2 1 + (-2.39 - 2.39i)T + 19iT^{2}
23 1−3.39T+23T2 1 - 3.39T + 23T^{2}
29 1−6.57iT−29T2 1 - 6.57iT - 29T^{2}
31 1+(−0.386−0.386i)T+31iT2 1 + (-0.386 - 0.386i)T + 31iT^{2}
37 1+(5.93−5.93i)T−37iT2 1 + (5.93 - 5.93i)T - 37iT^{2}
41 1+(0.734−0.734i)T−41iT2 1 + (0.734 - 0.734i)T - 41iT^{2}
43 1+7.56iT−43T2 1 + 7.56iT - 43T^{2}
47 1+(−0.243−0.243i)T+47iT2 1 + (-0.243 - 0.243i)T + 47iT^{2}
53 1−2.07iT−53T2 1 - 2.07iT - 53T^{2}
59 1+(3.56+3.56i)T+59iT2 1 + (3.56 + 3.56i)T + 59iT^{2}
61 1−7.04T+61T2 1 - 7.04T + 61T^{2}
67 1+(4.54+4.54i)T+67iT2 1 + (4.54 + 4.54i)T + 67iT^{2}
71 1+(6.79−6.79i)T−71iT2 1 + (6.79 - 6.79i)T - 71iT^{2}
73 1+(6.04−6.04i)T−73iT2 1 + (6.04 - 6.04i)T - 73iT^{2}
79 1−8.77T+79T2 1 - 8.77T + 79T^{2}
83 1+(8.31−8.31i)T−83iT2 1 + (8.31 - 8.31i)T - 83iT^{2}
89 1+(−9.62−9.62i)T+89iT2 1 + (-9.62 - 9.62i)T + 89iT^{2}
97 1+(1.34+1.34i)T+97iT2 1 + (1.34 + 1.34i)T + 97iT^{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

−11.29390674669068407003711423071, −10.42954171842563566890690370184, −9.358352845990354144613329835879, −8.861967121882180784915834205789, −8.252593429284326970881783114104, −6.89569146987554728763792258309, −5.79828134939649191562905455335, −5.00352332391810949045483478786, −3.53286573605373569119402192896, −2.75237694607163648680288943024, 0.30164150988452616019462070827, 2.00437191366937395691203114452, 2.80537994486316409446654022527, 4.74309536673822725317099114750, 5.95040946932368843697011168476, 6.76353766429349568519596496886, 7.44962069425104202142047611890, 8.730711590437490341167478019563, 9.633854964534891780794601415612, 10.38728113710160771827157830801

Graph of the ZZ-function along the critical line