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Properties

Label 2-475-5.4-c1-0-18
Degree 22
Conductor 475475
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 3.792893.79289
Root an. cond. 1.947531.94753
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  − 1.48i·2-s − 0.806i·3-s − 0.193·4-s − 1.19·6-s + 3.35i·7-s − 2.67i·8-s + 2.35·9-s + 0.962·11-s + 0.156i·12-s − 6.15i·13-s + 4.96·14-s − 4.35·16-s − 6.31i·17-s − 3.48i·18-s + 19-s + ⋯
L(s)  = 1  − 1.04i·2-s − 0.465i·3-s − 0.0969·4-s − 0.487·6-s + 1.26i·7-s − 0.945i·8-s + 0.783·9-s + 0.290·11-s + 0.0451i·12-s − 1.70i·13-s + 1.32·14-s − 1.08·16-s − 1.53i·17-s − 0.820i·18-s + 0.229·19-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 3.792893.79289
Root analytic conductor: 1.947531.94753
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ475(324,⋅)\chi_{475} (324, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 475, ( :1/2), −0.447+0.894i)(2,\ 475,\ (\ :1/2),\ -0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.860484−1.39229i0.860484 - 1.39229i
L(12)L(\frac12) ≈\approx 0.860484−1.39229i0.860484 - 1.39229i
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)
bad5 1 1
19 1−T 1 - T
good2 1+1.48iT−2T2 1 + 1.48iT - 2T^{2}
3 1+0.806iT−3T2 1 + 0.806iT - 3T^{2}
7 1−3.35iT−7T2 1 - 3.35iT - 7T^{2}
11 1−0.962T+11T2 1 - 0.962T + 11T^{2}
13 1+6.15iT−13T2 1 + 6.15iT - 13T^{2}
17 1+6.31iT−17T2 1 + 6.31iT - 17T^{2}
23 1−4.96iT−23T2 1 - 4.96iT - 23T^{2}
29 1−3.61T+29T2 1 - 3.61T + 29T^{2}
31 1+5.92T+31T2 1 + 5.92T + 31T^{2}
37 1−10.1iT−37T2 1 - 10.1iT - 37T^{2}
41 1−6.31T+41T2 1 - 6.31T + 41T^{2}
43 1−4.12iT−43T2 1 - 4.12iT - 43T^{2}
47 1−3.35iT−47T2 1 - 3.35iT - 47T^{2}
53 1+1.84iT−53T2 1 + 1.84iT - 53T^{2}
59 1−6.38T+59T2 1 - 6.38T + 59T^{2}
61 1+11.2T+61T2 1 + 11.2T + 61T^{2}
67 1+6.73iT−67T2 1 + 6.73iT - 67T^{2}
71 1+0.775T+71T2 1 + 0.775T + 71T^{2}
73 1+0.387iT−73T2 1 + 0.387iT - 73T^{2}
79 1−0.836T+79T2 1 - 0.836T + 79T^{2}
83 1−7.03iT−83T2 1 - 7.03iT - 83T^{2}
89 1+7.08T+89T2 1 + 7.08T + 89T^{2}
97 1−10.9iT−97T2 1 - 10.9iT - 97T^{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

−10.87330483658798523731711821941, −9.840112875962938741536600039919, −9.304425900635043248687814289381, −7.979247868708084433214581644409, −7.11773524871055099062767365417, −5.99240968010511142263142228909, −4.91071574355800502009757620631, −3.30449075657326734063347055999, −2.49977397895818048729073091438, −1.13577466378416316519171220224, 1.79980934376102832186888373327, 3.97165143462199310648785119715, 4.45394485049696642444098347691, 5.91032526669856359768868508896, 6.91051756611928927108608369606, 7.28939368428451597780305742248, 8.491244452330510267636194237337, 9.374241071299322481098771766690, 10.52087605087791118043096793097, 10.96793069604540662808276854664

Graph of the ZZ-function along the critical line