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Properties

Label 2-475-19.11-c1-0-14
Degree 22
Conductor 475475
Sign 0.981−0.189i0.981 - 0.189i
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 + (−1.47 + 2.55i)3-s + (−1.35 − 2.34i)4-s + (−3.20 − 5.54i)6-s + 0.591·7-s + 1.53·8-s + (−2.85 − 4.94i)9-s + 2.58·11-s + 7.99·12-s + (−3.43 − 5.94i)13-s + (−0.641 + 1.11i)14-s + (1.03 − 1.79i)16-s + (2.61 − 4.53i)17-s + 12.3·18-s + (−2.26 + 3.72i)19-s + ⋯
L(s)  = 1  + (−0.767 + 1.32i)2-s + (−0.852 + 1.47i)3-s + (−0.677 − 1.17i)4-s + (−1.30 − 2.26i)6-s + 0.223·7-s + 0.544·8-s + (−0.952 − 1.64i)9-s + 0.778·11-s + 2.30·12-s + (−0.952 − 1.64i)13-s + (−0.171 + 0.297i)14-s + (0.259 − 0.449i)16-s + (0.634 − 1.09i)17-s + 2.92·18-s + (−0.519 + 0.854i)19-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((0.981−0.189i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((0.981−0.189i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: 0.981−0.189i0.981 - 0.189i
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), 0.981−0.189i)(2,\ 475,\ (\ :1/2),\ 0.981 - 0.189i)

Particular Values

L(1)L(1) ≈\approx 0.233514+0.0222797i0.233514 + 0.0222797i
L(12)L(\frac12) ≈\approx 0.233514+0.0222797i0.233514 + 0.0222797i
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+(2.26−3.72i)T 1 + (2.26 - 3.72i)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+(1.47−2.55i)T+(−1.5−2.59i)T2 1 + (1.47 - 2.55i)T + (-1.5 - 2.59i)T^{2}
7 1−0.591T+7T2 1 - 0.591T + 7T^{2}
11 1−2.58T+11T2 1 - 2.58T + 11T^{2}
13 1+(3.43+5.94i)T+(−6.5+11.2i)T2 1 + (3.43 + 5.94i)T + (-6.5 + 11.2i)T^{2}
17 1+(−2.61+4.53i)T+(−8.5−14.7i)T2 1 + (-2.61 + 4.53i)T + (-8.5 - 14.7i)T^{2}
23 1+(1.45+2.51i)T+(−11.5+19.9i)T2 1 + (1.45 + 2.51i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.52+6.10i)T+(−14.5+25.1i)T2 1 + (3.52 + 6.10i)T + (-14.5 + 25.1i)T^{2}
31 1+6.81T+31T2 1 + 6.81T + 31T^{2}
37 1+4.82T+37T2 1 + 4.82T + 37T^{2}
41 1+(3.11−5.39i)T+(−20.5−35.5i)T2 1 + (3.11 - 5.39i)T + (-20.5 - 35.5i)T^{2}
43 1+(2.18−3.77i)T+(−21.5−37.2i)T2 1 + (2.18 - 3.77i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.27+2.21i)T+(−23.5+40.7i)T2 1 + (1.27 + 2.21i)T + (-23.5 + 40.7i)T^{2}
53 1+(−4.79−8.30i)T+(−26.5+45.8i)T2 1 + (-4.79 - 8.30i)T + (-26.5 + 45.8i)T^{2}
59 1+(−1.46+2.53i)T+(−29.5−51.0i)T2 1 + (-1.46 + 2.53i)T + (-29.5 - 51.0i)T^{2}
61 1+(1.16+2.01i)T+(−30.5+52.8i)T2 1 + (1.16 + 2.01i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.15+3.72i)T+(−33.5+58.0i)T2 1 + (2.15 + 3.72i)T + (-33.5 + 58.0i)T^{2}
71 1+(−6.74+11.6i)T+(−35.5−61.4i)T2 1 + (-6.74 + 11.6i)T + (-35.5 - 61.4i)T^{2}
73 1+(−4.21+7.29i)T+(−36.5−63.2i)T2 1 + (-4.21 + 7.29i)T + (-36.5 - 63.2i)T^{2}
79 1+(2.93−5.08i)T+(−39.5−68.4i)T2 1 + (2.93 - 5.08i)T + (-39.5 - 68.4i)T^{2}
83 1+4.02T+83T2 1 + 4.02T + 83T^{2}
89 1+(1.85+3.21i)T+(−44.5+77.0i)T2 1 + (1.85 + 3.21i)T + (-44.5 + 77.0i)T^{2}
97 1+(1.26−2.18i)T+(−48.5−84.0i)T2 1 + (1.26 - 2.18i)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

−10.63994572131221698983166694469, −9.851730236644133317268565437849, −9.447755446700107801281524442352, −8.292029889723161211591547786462, −7.45507788860271529393423490597, −6.22714176789876102204113146462, −5.49669475375315674111875247463, −4.77729724064049683418017150367, −3.39168940184628059638862076542, −0.20833379265042068267002087974, 1.51460073396632903357543777364, 2.05755558609525390689132929757, 3.82899261827583705117741769017, 5.39900321785720619239792652892, 6.64301845586023668846479348419, 7.25614137920916750281528064427, 8.490253122290367439964315343501, 9.245136516241329228820814902086, 10.33780934533785417938834378679, 11.35071469558657578430102098175

Graph of the ZZ-function along the critical line