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Properties

Label 2-475-19.11-c1-0-15
Degree 22
Conductor 475475
Sign 0.655+0.755i0.655 + 0.755i
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  + (0.740 − 1.28i)2-s + (0.0908 − 0.157i)3-s + (−0.0969 − 0.167i)4-s + (−0.134 − 0.232i)6-s − 1.30·7-s + 2.67·8-s + (1.48 + 2.56i)9-s + 4.98·11-s − 0.0352·12-s + (0.203 + 0.351i)13-s + (−0.965 + 1.67i)14-s + (2.17 − 3.76i)16-s + (1.37 − 2.38i)17-s + 4.39·18-s + (−4.35 − 0.0955i)19-s + ⋯
L(s)  = 1  + (0.523 − 0.907i)2-s + (0.0524 − 0.0908i)3-s + (−0.0484 − 0.0839i)4-s + (−0.0549 − 0.0951i)6-s − 0.492·7-s + 0.945·8-s + (0.494 + 0.856i)9-s + 1.50·11-s − 0.0101·12-s + (0.0563 + 0.0975i)13-s + (−0.258 + 0.447i)14-s + (0.543 − 0.941i)16-s + (0.333 − 0.577i)17-s + 1.03·18-s + (−0.999 − 0.0219i)19-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((0.655+0.755i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((0.655+0.755i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: 0.655+0.755i0.655 + 0.755i
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), 0.655+0.755i)(2,\ 475,\ (\ :1/2),\ 0.655 + 0.755i)

Particular Values

L(1)L(1) ≈\approx 1.94452−0.887400i1.94452 - 0.887400i
L(12)L(\frac12) ≈\approx 1.94452−0.887400i1.94452 - 0.887400i
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+(4.35+0.0955i)T 1 + (4.35 + 0.0955i)T
good2 1+(−0.740+1.28i)T+(−1−1.73i)T2 1 + (-0.740 + 1.28i)T + (-1 - 1.73i)T^{2}
3 1+(−0.0908+0.157i)T+(−1.5−2.59i)T2 1 + (-0.0908 + 0.157i)T + (-1.5 - 2.59i)T^{2}
7 1+1.30T+7T2 1 + 1.30T + 7T^{2}
11 1−4.98T+11T2 1 - 4.98T + 11T^{2}
13 1+(−0.203−0.351i)T+(−6.5+11.2i)T2 1 + (-0.203 - 0.351i)T + (-6.5 + 11.2i)T^{2}
17 1+(−1.37+2.38i)T+(−8.5−14.7i)T2 1 + (-1.37 + 2.38i)T + (-8.5 - 14.7i)T^{2}
23 1+(3.47+6.02i)T+(−11.5+19.9i)T2 1 + (3.47 + 6.02i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.00−3.47i)T+(−14.5+25.1i)T2 1 + (-2.00 - 3.47i)T + (-14.5 + 25.1i)T^{2}
31 1+2.57T+31T2 1 + 2.57T + 31T^{2}
37 1−3.71T+37T2 1 - 3.71T + 37T^{2}
41 1+(−0.607+1.05i)T+(−20.5−35.5i)T2 1 + (-0.607 + 1.05i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.56−2.70i)T+(−21.5−37.2i)T2 1 + (1.56 - 2.70i)T + (-21.5 - 37.2i)T^{2}
47 1+(−3.25−5.63i)T+(−23.5+40.7i)T2 1 + (-3.25 - 5.63i)T + (-23.5 + 40.7i)T^{2}
53 1+(3.16+5.47i)T+(−26.5+45.8i)T2 1 + (3.16 + 5.47i)T + (-26.5 + 45.8i)T^{2}
59 1+(5.61−9.72i)T+(−29.5−51.0i)T2 1 + (5.61 - 9.72i)T + (-29.5 - 51.0i)T^{2}
61 1+(0.467+0.808i)T+(−30.5+52.8i)T2 1 + (0.467 + 0.808i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.64+4.58i)T+(−33.5+58.0i)T2 1 + (2.64 + 4.58i)T + (-33.5 + 58.0i)T^{2}
71 1+(−0.817+1.41i)T+(−35.5−61.4i)T2 1 + (-0.817 + 1.41i)T + (-35.5 - 61.4i)T^{2}
73 1+(3.84−6.65i)T+(−36.5−63.2i)T2 1 + (3.84 - 6.65i)T + (-36.5 - 63.2i)T^{2}
79 1+(−7.27+12.6i)T+(−39.5−68.4i)T2 1 + (-7.27 + 12.6i)T + (-39.5 - 68.4i)T^{2}
83 1+15.2T+83T2 1 + 15.2T + 83T^{2}
89 1+(7.10+12.3i)T+(−44.5+77.0i)T2 1 + (7.10 + 12.3i)T + (-44.5 + 77.0i)T^{2}
97 1+(9.14−15.8i)T+(−48.5−84.0i)T2 1 + (9.14 - 15.8i)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

−10.96580965636596278950322980409, −10.27180344400938729626010756896, −9.304593322032336632229757201818, −8.195022561223160123966831125031, −7.13873966014677926655071741802, −6.27311682514988582592674196787, −4.69297010501616871105268148245, −4.00099082697031471097229663204, −2.77101835865932690081705794350, −1.57318216110760876594210483812, 1.52734987891438424828430035882, 3.66835510561360995180920661237, 4.31865729961090625826213148827, 5.82557199429568837331335016103, 6.39624724836649779889770172940, 7.13053286013700905420601326187, 8.266371119638830772881463038706, 9.404404931488522472359316983062, 10.03936143948877775455808582500, 11.18057284716157059918194555135

Graph of the ZZ-function along the critical line