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Properties

Label 2-475-19.11-c1-0-11
Degree 22
Conductor 475475
Sign −0.0977−0.995i-0.0977 - 0.995i
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 + 1.73i)3-s + (1 + 1.73i)4-s + 4·7-s + (−0.499 − 0.866i)9-s + 3·11-s − 3.99·12-s + (1 + 1.73i)13-s + (−1.99 + 3.46i)16-s + (3 − 5.19i)17-s + (−3.5 − 2.59i)19-s + (−4 + 6.92i)21-s − 4.00·27-s + (4 + 6.92i)28-s + (1.5 + 2.59i)29-s − 7·31-s + ⋯
L(s)  = 1  + (−0.577 + 0.999i)3-s + (0.5 + 0.866i)4-s + 1.51·7-s + (−0.166 − 0.288i)9-s + 0.904·11-s − 1.15·12-s + (0.277 + 0.480i)13-s + (−0.499 + 0.866i)16-s + (0.727 − 1.26i)17-s + (−0.802 − 0.596i)19-s + (−0.872 + 1.51i)21-s − 0.769·27-s + (0.755 + 1.30i)28-s + (0.278 + 0.482i)29-s − 1.25·31-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((−0.0977−0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((−0.0977−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: −0.0977−0.995i-0.0977 - 0.995i
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), −0.0977−0.995i)(2,\ 475,\ (\ :1/2),\ -0.0977 - 0.995i)

Particular Values

L(1)L(1) ≈\approx 1.06118+1.17050i1.06118 + 1.17050i
L(12)L(\frac12) ≈\approx 1.06118+1.17050i1.06118 + 1.17050i
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+(3.5+2.59i)T 1 + (3.5 + 2.59i)T
good2 1+(−1−1.73i)T2 1 + (-1 - 1.73i)T^{2}
3 1+(1−1.73i)T+(−1.5−2.59i)T2 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2}
7 1−4T+7T2 1 - 4T + 7T^{2}
11 1−3T+11T2 1 - 3T + 11T^{2}
13 1+(−1−1.73i)T+(−6.5+11.2i)T2 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2}
17 1+(−3+5.19i)T+(−8.5−14.7i)T2 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1+(−1.5−2.59i)T+(−14.5+25.1i)T2 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2}
31 1+7T+31T2 1 + 7T + 31T^{2}
37 1+8T+37T2 1 + 8T + 37T^{2}
41 1+(−3+5.19i)T+(−20.5−35.5i)T2 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2}
43 1+(2−3.46i)T+(−21.5−37.2i)T2 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2}
47 1+(−3−5.19i)T+(−23.5+40.7i)T2 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2}
53 1+(3+5.19i)T+(−26.5+45.8i)T2 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2}
59 1+(−7.5+12.9i)T+(−29.5−51.0i)T2 1 + (-7.5 + 12.9i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.5+4.33i)T+(−30.5+52.8i)T2 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1−1.73i)T+(−33.5+58.0i)T2 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2}
71 1+(−1.5+2.59i)T+(−35.5−61.4i)T2 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2}
73 1+(−4+6.92i)T+(−36.5−63.2i)T2 1 + (-4 + 6.92i)T + (-36.5 - 63.2i)T^{2}
79 1+(2.5−4.33i)T+(−39.5−68.4i)T2 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2}
83 1+12T+83T2 1 + 12T + 83T^{2}
89 1+(−7.5−12.9i)T+(−44.5+77.0i)T2 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2}
97 1+(−4+6.92i)T+(−48.5−84.0i)T2 1 + (-4 + 6.92i)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

−11.25680112181617328687651782649, −10.71593279964040892144337013764, −9.412174090981647598879264232348, −8.614350793160160309967491943006, −7.60332834659405498782401041705, −6.72053709103147934181714728347, −5.30231486765980986190015992088, −4.53081813332288507663557053442, −3.59627781104575035677850307993, −1.90450901963427336876409659814, 1.24096142123401968690163984206, 1.86919817563922856821254525608, 4.03663261716282539335541559312, 5.45898555686003375075929085438, 6.03635348207729154368808649323, 6.99493743745264066898582476958, 7.88098328543309962338861424988, 8.812319320166885670974834814309, 10.23460724456847848843181183484, 10.86667897450936964890459142585

Graph of the ZZ-function along the critical line