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Properties

Label 2-475-19.11-c1-0-16
Degree 22
Conductor 475475
Sign 0.332+0.943i0.332 + 0.943i
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  + (1.08 − 1.87i)2-s + (−0.706 + 1.22i)3-s + (−1.35 − 2.34i)4-s + (1.53 + 2.65i)6-s + 1.76·7-s − 1.53·8-s + (0.502 + 0.869i)9-s + 1.83·11-s + 3.82·12-s + (−1.30 − 2.25i)13-s + (1.91 − 3.31i)14-s + (1.03 − 1.79i)16-s + (2.11 − 3.66i)17-s + 2.17·18-s + (4.01 + 1.68i)19-s + ⋯
L(s)  = 1  + (0.767 − 1.32i)2-s + (−0.407 + 0.706i)3-s + (−0.677 − 1.17i)4-s + (0.625 + 1.08i)6-s + 0.665·7-s − 0.544·8-s + (0.167 + 0.289i)9-s + 0.554·11-s + 1.10·12-s + (−0.361 − 0.625i)13-s + (0.510 − 0.884i)14-s + (0.259 − 0.449i)16-s + (0.513 − 0.889i)17-s + 0.513·18-s + (0.922 + 0.386i)19-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((0.332+0.943i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((0.332+0.943i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: 0.332+0.943i0.332 + 0.943i
Analytic conductor: 3.792893.79289
Root analytic conductor: 1.947531.94753
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ475(201,⋅)\chi_{475} (201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 475, ( :1/2), 0.332+0.943i)(2,\ 475,\ (\ :1/2),\ 0.332 + 0.943i)

Particular Values

L(1)L(1) ≈\approx 1.68206−1.19028i1.68206 - 1.19028i
L(12)L(\frac12) ≈\approx 1.68206−1.19028i1.68206 - 1.19028i
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+(−4.01−1.68i)T 1 + (-4.01 - 1.68i)T
good2 1+(−1.08+1.87i)T+(−1−1.73i)T2 1 + (-1.08 + 1.87i)T + (-1 - 1.73i)T^{2}
3 1+(0.706−1.22i)T+(−1.5−2.59i)T2 1 + (0.706 - 1.22i)T + (-1.5 - 2.59i)T^{2}
7 1−1.76T+7T2 1 - 1.76T + 7T^{2}
11 1−1.83T+11T2 1 - 1.83T + 11T^{2}
13 1+(1.30+2.25i)T+(−6.5+11.2i)T2 1 + (1.30 + 2.25i)T + (-6.5 + 11.2i)T^{2}
17 1+(−2.11+3.66i)T+(−8.5−14.7i)T2 1 + (-2.11 + 3.66i)T + (-8.5 - 14.7i)T^{2}
23 1+(1.10+1.91i)T+(−11.5+19.9i)T2 1 + (1.10 + 1.91i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.56−6.17i)T+(−14.5+25.1i)T2 1 + (-3.56 - 6.17i)T + (-14.5 + 25.1i)T^{2}
31 1−0.303T+31T2 1 - 0.303T + 31T^{2}
37 1+3.90T+37T2 1 + 3.90T + 37T^{2}
41 1+(4.11−7.13i)T+(−20.5−35.5i)T2 1 + (4.11 - 7.13i)T + (-20.5 - 35.5i)T^{2}
43 1+(−1.17+2.03i)T+(−21.5−37.2i)T2 1 + (-1.17 + 2.03i)T + (-21.5 - 37.2i)T^{2}
47 1+(3.62+6.28i)T+(−23.5+40.7i)T2 1 + (3.62 + 6.28i)T + (-23.5 + 40.7i)T^{2}
53 1+(−5.31−9.19i)T+(−26.5+45.8i)T2 1 + (-5.31 - 9.19i)T + (-26.5 + 45.8i)T^{2}
59 1+(−6.02+10.4i)T+(−29.5−51.0i)T2 1 + (-6.02 + 10.4i)T + (-29.5 - 51.0i)T^{2}
61 1+(5.26+9.12i)T+(−30.5+52.8i)T2 1 + (5.26 + 9.12i)T + (-30.5 + 52.8i)T^{2}
67 1+(−6.51−11.2i)T+(−33.5+58.0i)T2 1 + (-6.51 - 11.2i)T + (-33.5 + 58.0i)T^{2}
71 1+(5.91−10.2i)T+(−35.5−61.4i)T2 1 + (5.91 - 10.2i)T + (-35.5 - 61.4i)T^{2}
73 1+(4.58−7.94i)T+(−36.5−63.2i)T2 1 + (4.58 - 7.94i)T + (-36.5 - 63.2i)T^{2}
79 1+(3.94−6.82i)T+(−39.5−68.4i)T2 1 + (3.94 - 6.82i)T + (-39.5 - 68.4i)T^{2}
83 1+6.93T+83T2 1 + 6.93T + 83T^{2}
89 1+(−6.23−10.8i)T+(−44.5+77.0i)T2 1 + (-6.23 - 10.8i)T + (-44.5 + 77.0i)T^{2}
97 1+(−3.87+6.71i)T+(−48.5−84.0i)T2 1 + (-3.87 + 6.71i)T + (-48.5 - 84.0i)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

−10.99989889606092394000058552805, −10.14442306589116038060584393285, −9.684881741262587195575539495634, −8.249870683064506534727862003919, −7.13148562562567071885866991758, −5.37830943326653019618461497525, −4.98331111268418229389663113995, −3.94647700606612471669230691532, −2.87291219605138483216826201347, −1.39060147000619639032962851981, 1.55624551876429758187879874596, 3.73892980984168203993003651787, 4.74526988216744419622245001623, 5.77177282737129438485780330387, 6.50441593343956871567148406917, 7.30562440121038654567632822860, 7.979300518497352476717768565332, 9.092044827077187251482868267804, 10.29104767220110386807721191703, 11.68950967558571513091075038475

Graph of the ZZ-function along the critical line