↑

Properties

Label 2-475-19.11-c1-0-12
Degree 22
Conductor 475475
Sign 0.910−0.412i0.910 - 0.412i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.25 + 2.17i)2-s + (0.610 − 1.05i)3-s + (−2.14 − 3.71i)4-s + (1.53 + 2.65i)6-s + 0.221·7-s + 5.72·8-s + (0.753 + 1.30i)9-s − 0.778·11-s − 5.23·12-s + (−2.5 − 4.33i)13-s + (−0.278 + 0.481i)14-s + (−2.89 + 5.01i)16-s + (3.53 − 6.12i)17-s − 3.77·18-s + (1.33 − 4.15i)19-s + ⋯
L(s)  = 1  + (−0.886 + 1.53i)2-s + (0.352 − 0.610i)3-s + (−1.07 − 1.85i)4-s + (0.625 + 1.08i)6-s + 0.0838·7-s + 2.02·8-s + (0.251 + 0.435i)9-s − 0.234·11-s − 1.51·12-s + (−0.693 − 1.20i)13-s + (−0.0743 + 0.128i)14-s + (−0.724 + 1.25i)16-s + (0.858 − 1.48i)17-s − 0.890·18-s + (0.305 − 0.952i)19-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((0.910−0.412i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((0.910−0.412i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: 0.910−0.412i0.910 - 0.412i
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), 0.910−0.412i)(2,\ 475,\ (\ :1/2),\ 0.910 - 0.412i)

Particular Values

L(1)L(1) ≈\approx 0.904484+0.195483i0.904484 + 0.195483i
L(12)L(\frac12) ≈\approx 0.904484+0.195483i0.904484 + 0.195483i
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+(−1.33+4.15i)T 1 + (-1.33 + 4.15i)T
good2 1+(1.25−2.17i)T+(−1−1.73i)T2 1 + (1.25 - 2.17i)T + (-1 - 1.73i)T^{2}
3 1+(−0.610+1.05i)T+(−1.5−2.59i)T2 1 + (-0.610 + 1.05i)T + (-1.5 - 2.59i)T^{2}
7 1−0.221T+7T2 1 - 0.221T + 7T^{2}
11 1+0.778T+11T2 1 + 0.778T + 11T^{2}
13 1+(2.5+4.33i)T+(−6.5+11.2i)T2 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2}
17 1+(−3.53+6.12i)T+(−8.5−14.7i)T2 1 + (-3.53 + 6.12i)T + (-8.5 - 14.7i)T^{2}
23 1+(−4.03−6.99i)T+(−11.5+19.9i)T2 1 + (-4.03 - 6.99i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.110+0.192i)T+(−14.5+25.1i)T2 1 + (0.110 + 0.192i)T + (-14.5 + 25.1i)T^{2}
31 1−2.50T+31T2 1 - 2.50T + 31T^{2}
37 1−1.90T+37T2 1 - 1.90T + 37T^{2}
41 1+(−3.61+6.26i)T+(−20.5−35.5i)T2 1 + (-3.61 + 6.26i)T + (-20.5 - 35.5i)T^{2}
43 1+(−3.64+6.32i)T+(−21.5−37.2i)T2 1 + (-3.64 + 6.32i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.39+2.41i)T+(−23.5+40.7i)T2 1 + (1.39 + 2.41i)T + (-23.5 + 40.7i)T^{2}
53 1+(−2.19−3.79i)T+(−26.5+45.8i)T2 1 + (-2.19 - 3.79i)T + (-26.5 + 45.8i)T^{2}
59 1+(−1.39+2.41i)T+(−29.5−51.0i)T2 1 + (-1.39 + 2.41i)T + (-29.5 - 51.0i)T^{2}
61 1+(−6.29−10.8i)T+(−30.5+52.8i)T2 1 + (-6.29 - 10.8i)T + (-30.5 + 52.8i)T^{2}
67 1+(5.28+9.15i)T+(−33.5+58.0i)T2 1 + (5.28 + 9.15i)T + (-33.5 + 58.0i)T^{2}
71 1+(−4.92+8.52i)T+(−35.5−61.4i)T2 1 + (-4.92 + 8.52i)T + (-35.5 - 61.4i)T^{2}
73 1+(7.03−12.1i)T+(−36.5−63.2i)T2 1 + (7.03 - 12.1i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.792−1.37i)T+(−39.5−68.4i)T2 1 + (0.792 - 1.37i)T + (-39.5 - 68.4i)T^{2}
83 1−9.52T+83T2 1 - 9.52T + 83T^{2}
89 1+(1.57+2.71i)T+(−44.5+77.0i)T2 1 + (1.57 + 2.71i)T + (-44.5 + 77.0i)T^{2}
97 1+(3.18−5.51i)T+(−48.5−84.0i)T2 1 + (3.18 - 5.51i)T + (-48.5 - 84.0i)T^{2}
show more
show less
   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

−10.69031879008068018147244449966, −9.778287908915256567996642701861, −9.085824456466484498327632318087, −7.984010942388622338837020732399, −7.45256236702996570759877284254, −6.98806444053600022878436202671, −5.49100037982058781935466097656, −5.02710603811206847707130621460, −2.80181047719129562755059651019, −0.851493773775849930987144135948, 1.37590370879508833567984685784, 2.72906864628016150800411087153, 3.79261435691656696779702123729, 4.59389301000982668979591950169, 6.41999319106522427040436988799, 7.86060123988430621937283087617, 8.606375649925596022542469409066, 9.506449086445786822578826774308, 10.00698133303412958156068315149, 10.72570370520519314144769957306

Graph of the ZZ-function along the critical line