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Properties

Label 2-507-39.8-c1-0-30
Degree 22
Conductor 507507
Sign 0.0145+0.999i0.0145 + 0.999i
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  + (−1.42 + 1.42i)2-s + (−1.57 − 0.730i)3-s − 2.07i·4-s + (1.72 − 1.72i)5-s + (3.28 − 1.19i)6-s + (2.20 − 2.20i)7-s + (0.105 + 0.105i)8-s + (1.93 + 2.29i)9-s + 4.91i·10-s + (−1.95 − 1.95i)11-s + (−1.51 + 3.25i)12-s + 6.28i·14-s + (−3.96 + 1.44i)15-s + 3.84·16-s − 5.78·17-s + (−6.03 − 0.515i)18-s + ⋯
L(s)  = 1  + (−1.00 + 1.00i)2-s + (−0.906 − 0.421i)3-s − 1.03i·4-s + (0.770 − 0.770i)5-s + (1.34 − 0.489i)6-s + (0.831 − 0.831i)7-s + (0.0372 + 0.0372i)8-s + (0.644 + 0.764i)9-s + 1.55i·10-s + (−0.590 − 0.590i)11-s + (−0.437 + 0.940i)12-s + 1.67i·14-s + (−1.02 + 0.373i)15-s + 0.961·16-s − 1.40·17-s + (−1.42 − 0.121i)18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.0145+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0145 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((0.0145+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0145 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.0145+0.999i0.0145 + 0.999i
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), 0.0145+0.999i)(2,\ 507,\ (\ :1/2),\ 0.0145 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.328734−0.323969i0.328734 - 0.323969i
L(12)L(\frac12) ≈\approx 0.328734−0.323969i0.328734 - 0.323969i
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+(1.57+0.730i)T 1 + (1.57 + 0.730i)T
13 1 1
good2 1+(1.42−1.42i)T−2iT2 1 + (1.42 - 1.42i)T - 2iT^{2}
5 1+(−1.72+1.72i)T−5iT2 1 + (-1.72 + 1.72i)T - 5iT^{2}
7 1+(−2.20+2.20i)T−7iT2 1 + (-2.20 + 2.20i)T - 7iT^{2}
11 1+(1.95+1.95i)T+11iT2 1 + (1.95 + 1.95i)T + 11iT^{2}
17 1+5.78T+17T2 1 + 5.78T + 17T^{2}
19 1+(1.06+1.06i)T+19iT2 1 + (1.06 + 1.06i)T + 19iT^{2}
23 1+3.86T+23T2 1 + 3.86T + 23T^{2}
29 1−2.92iT−29T2 1 - 2.92iT - 29T^{2}
31 1+(3.56+3.56i)T+31iT2 1 + (3.56 + 3.56i)T + 31iT^{2}
37 1+(−2.83+2.83i)T−37iT2 1 + (-2.83 + 2.83i)T - 37iT^{2}
41 1+(−4.79+4.79i)T−41iT2 1 + (-4.79 + 4.79i)T - 41iT^{2}
43 1−1.84iT−43T2 1 - 1.84iT - 43T^{2}
47 1+(−0.115−0.115i)T+47iT2 1 + (-0.115 - 0.115i)T + 47iT^{2}
53 1+10.0iT−53T2 1 + 10.0iT - 53T^{2}
59 1+(1.22+1.22i)T+59iT2 1 + (1.22 + 1.22i)T + 59iT^{2}
61 1+5.39T+61T2 1 + 5.39T + 61T^{2}
67 1+(9.65+9.65i)T+67iT2 1 + (9.65 + 9.65i)T + 67iT^{2}
71 1+(0.239−0.239i)T−71iT2 1 + (0.239 - 0.239i)T - 71iT^{2}
73 1+(8.54−8.54i)T−73iT2 1 + (8.54 - 8.54i)T - 73iT^{2}
79 1−10.4T+79T2 1 - 10.4T + 79T^{2}
83 1+(−2.83+2.83i)T−83iT2 1 + (-2.83 + 2.83i)T - 83iT^{2}
89 1+(−3.00−3.00i)T+89iT2 1 + (-3.00 - 3.00i)T + 89iT^{2}
97 1+(6.99+6.99i)T+97iT2 1 + (6.99 + 6.99i)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

−10.67069578915175867253752703762, −9.636408120232575363680695167022, −8.733579313362687662688405364431, −7.899941935700200335353720479503, −7.15531129566005872930949563341, −6.17157994583765956409065302635, −5.42845400978754467046254237867, −4.39779052179380648687511603467, −1.78067541160611119220414621616, −0.42073744241789426135081391117, 1.79113811431017352252476764307, 2.64438751125992213647916930452, 4.43336770451677560977975615028, 5.59502183345908268390448849414, 6.40557452476354963541770319709, 7.75062963446785471150068928539, 8.884257893180446111841019160259, 9.598490055641155631611235548226, 10.48891496299737237913382887208, 10.80611093920414794995824449297

Graph of the ZZ-function along the critical line