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Properties

Label 2-507-39.5-c1-0-10
Degree 22
Conductor 507507
Sign −0.980−0.198i-0.980 - 0.198i
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  + (1.69 + 1.69i)2-s + (−0.366 + 1.69i)3-s + 3.73i·4-s + (1.69 + 1.69i)5-s + (−3.48 + 2.24i)6-s + (−1 − i)7-s + (−2.93 + 2.93i)8-s + (−2.73 − 1.23i)9-s + 5.73i·10-s + (1.23 − 1.23i)11-s + (−6.31 − 1.36i)12-s − 3.38i·14-s + (−3.48 + 2.24i)15-s − 2.46·16-s + 2.14·17-s + (−2.52 − 6.72i)18-s + ⋯
L(s)  = 1  + (1.19 + 1.19i)2-s + (−0.211 + 0.977i)3-s + 1.86i·4-s + (0.757 + 0.757i)5-s + (−1.42 + 0.917i)6-s + (−0.377 − 0.377i)7-s + (−1.03 + 1.03i)8-s + (−0.910 − 0.413i)9-s + 1.81i·10-s + (0.373 − 0.373i)11-s + (−1.82 − 0.394i)12-s − 0.904i·14-s + (−0.899 + 0.580i)15-s − 0.616·16-s + 0.520·17-s + (−0.595 − 1.58i)18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.980−0.198i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.980−0.198i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.980−0.198i-0.980 - 0.198i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(239,⋅)\chi_{507} (239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.980−0.198i)(2,\ 507,\ (\ :1/2),\ -0.980 - 0.198i)

Particular Values

L(1)L(1) ≈\approx 0.262972+2.62143i0.262972 + 2.62143i
L(12)L(\frac12) ≈\approx 0.262972+2.62143i0.262972 + 2.62143i
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+(0.366−1.69i)T 1 + (0.366 - 1.69i)T
13 1 1
good2 1+(−1.69−1.69i)T+2iT2 1 + (-1.69 - 1.69i)T + 2iT^{2}
5 1+(−1.69−1.69i)T+5iT2 1 + (-1.69 - 1.69i)T + 5iT^{2}
7 1+(1+i)T+7iT2 1 + (1 + i)T + 7iT^{2}
11 1+(−1.23+1.23i)T−11iT2 1 + (-1.23 + 1.23i)T - 11iT^{2}
17 1−2.14T+17T2 1 - 2.14T + 17T^{2}
19 1+(−0.732+0.732i)T−19iT2 1 + (-0.732 + 0.732i)T - 19iT^{2}
23 1+23T2 1 + 23T^{2}
29 1+5.53iT−29T2 1 + 5.53iT - 29T^{2}
31 1+(4.46−4.46i)T−31iT2 1 + (4.46 - 4.46i)T - 31iT^{2}
37 1+(−4.83−4.83i)T+37iT2 1 + (-4.83 - 4.83i)T + 37iT^{2}
41 1+(0.453+0.453i)T+41iT2 1 + (0.453 + 0.453i)T + 41iT^{2}
43 1+8.19iT−43T2 1 + 8.19iT - 43T^{2}
47 1+(6.77−6.77i)T−47iT2 1 + (6.77 - 6.77i)T - 47iT^{2}
53 1−4.62iT−53T2 1 - 4.62iT - 53T^{2}
59 1+(−3.38+3.38i)T−59iT2 1 + (-3.38 + 3.38i)T - 59iT^{2}
61 1+7T+61T2 1 + 7T + 61T^{2}
67 1+(6.19−6.19i)T−67iT2 1 + (6.19 - 6.19i)T - 67iT^{2}
71 1+(3.38+3.38i)T+71iT2 1 + (3.38 + 3.38i)T + 71iT^{2}
73 1+(−6.09−6.09i)T+73iT2 1 + (-6.09 - 6.09i)T + 73iT^{2}
79 1−2T+79T2 1 - 2T + 79T^{2}
83 1+(1.23+1.23i)T+83iT2 1 + (1.23 + 1.23i)T + 83iT^{2}
89 1+(−7.10+7.10i)T−89iT2 1 + (-7.10 + 7.10i)T - 89iT^{2}
97 1+(9.19−9.19i)T−97iT2 1 + (9.19 - 9.19i)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

−11.37301481682894960287201719178, −10.35834460691950569740070693658, −9.666571579510559445786253536440, −8.501862320002860941864392624401, −7.30727389013985837338323838891, −6.35643181333625865873594855432, −5.85614244559148914101319139811, −4.85107761994723456363776783621, −3.80138962565327556255201378034, −2.97219060780929974974974115437, 1.27522687636798636771967110223, 2.19243624944762608849094562519, 3.39080671197382563548933872231, 4.83577485719117528973645449524, 5.61646917153109894097654741854, 6.32112100644091403783122983196, 7.67226141347859155148868675635, 9.031350613132712580229712917305, 9.763061502531668235877935342003, 10.86725014741183431811988811400

Graph of the ZZ-function along the critical line