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Properties

Label 2-950-19.11-c1-0-27
Degree 22
Conductor 950950
Sign −0.634+0.772i-0.634 + 0.772i
Analytic cond. 7.585787.58578
Root an. cond. 2.754232.75423
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  + (−0.5 + 0.866i)2-s + (0.341 − 0.590i)3-s + (−0.499 − 0.866i)4-s + (0.341 + 0.590i)6-s − 0.317·7-s + 0.999·8-s + (1.26 + 2.19i)9-s − 4.31·11-s − 0.682·12-s + (−3.14 − 5.45i)13-s + (0.158 − 0.275i)14-s + (−0.5 + 0.866i)16-s + (0.0669 − 0.115i)17-s − 2.53·18-s + (−4.05 − 1.60i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.196 − 0.341i)3-s + (−0.249 − 0.433i)4-s + (0.139 + 0.241i)6-s − 0.120·7-s + 0.353·8-s + (0.422 + 0.731i)9-s − 1.29·11-s − 0.196·12-s + (−0.873 − 1.51i)13-s + (0.0424 − 0.0735i)14-s + (−0.125 + 0.216i)16-s + (0.0162 − 0.0281i)17-s − 0.597·18-s + (−0.929 − 0.367i)19-s + ⋯

Functional equation

Λ(s)=(950s/2 ΓC(s) L(s)=((−0.634+0.772i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(950s/2 ΓC(s+1/2) L(s)=((−0.634+0.772i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 950950    =    2⋅52⋅192 \cdot 5^{2} \cdot 19
Sign: −0.634+0.772i-0.634 + 0.772i
Analytic conductor: 7.585787.58578
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ950(201,⋅)\chi_{950} (201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 950, ( :1/2), −0.634+0.772i)(2,\ 950,\ (\ :1/2),\ -0.634 + 0.772i)

Particular Values

L(1)L(1) ≈\approx 0.140552−0.297299i0.140552 - 0.297299i
L(12)L(\frac12) ≈\approx 0.140552−0.297299i0.140552 - 0.297299i
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)
bad2 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
5 1 1
19 1+(4.05+1.60i)T 1 + (4.05 + 1.60i)T
good3 1+(−0.341+0.590i)T+(−1.5−2.59i)T2 1 + (-0.341 + 0.590i)T + (-1.5 - 2.59i)T^{2}
7 1+0.317T+7T2 1 + 0.317T + 7T^{2}
11 1+4.31T+11T2 1 + 4.31T + 11T^{2}
13 1+(3.14+5.45i)T+(−6.5+11.2i)T2 1 + (3.14 + 5.45i)T + (-6.5 + 11.2i)T^{2}
17 1+(−0.0669+0.115i)T+(−8.5−14.7i)T2 1 + (-0.0669 + 0.115i)T + (-8.5 - 14.7i)T^{2}
23 1+(1.98+3.44i)T+(−11.5+19.9i)T2 1 + (1.98 + 3.44i)T + (-11.5 + 19.9i)T^{2}
29 1+(−4.57−7.92i)T+(−14.5+25.1i)T2 1 + (-4.57 - 7.92i)T + (-14.5 + 25.1i)T^{2}
31 1+2.98T+31T2 1 + 2.98T + 31T^{2}
37 1+5.07T+37T2 1 + 5.07T + 37T^{2}
41 1+(−0.433+0.750i)T+(−20.5−35.5i)T2 1 + (-0.433 + 0.750i)T + (-20.5 - 35.5i)T^{2}
43 1+(2.85−4.93i)T+(−21.5−37.2i)T2 1 + (2.85 - 4.93i)T + (-21.5 - 37.2i)T^{2}
47 1+(6.48+11.2i)T+(−23.5+40.7i)T2 1 + (6.48 + 11.2i)T + (-23.5 + 40.7i)T^{2}
53 1+(3.96+6.86i)T+(−26.5+45.8i)T2 1 + (3.96 + 6.86i)T + (-26.5 + 45.8i)T^{2}
59 1+(−4.80+8.32i)T+(−29.5−51.0i)T2 1 + (-4.80 + 8.32i)T + (-29.5 - 51.0i)T^{2}
61 1+(3.08+5.34i)T+(−30.5+52.8i)T2 1 + (3.08 + 5.34i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.295−0.511i)T+(−33.5+58.0i)T2 1 + (-0.295 - 0.511i)T + (-33.5 + 58.0i)T^{2}
71 1+(5.83−10.0i)T+(−35.5−61.4i)T2 1 + (5.83 - 10.0i)T + (-35.5 - 61.4i)T^{2}
73 1+(−4.13+7.15i)T+(−36.5−63.2i)T2 1 + (-4.13 + 7.15i)T + (-36.5 - 63.2i)T^{2}
79 1+(−1.66+2.87i)T+(−39.5−68.4i)T2 1 + (-1.66 + 2.87i)T + (-39.5 - 68.4i)T^{2}
83 1+4.20T+83T2 1 + 4.20T + 83T^{2}
89 1+(−1.85−3.22i)T+(−44.5+77.0i)T2 1 + (-1.85 - 3.22i)T + (-44.5 + 77.0i)T^{2}
97 1+(2.42−4.20i)T+(−48.5−84.0i)T2 1 + (2.42 - 4.20i)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

−9.922798903916987538351296848998, −8.470201272741297006271829096682, −8.139592628966189906898101456115, −7.30310028343870608233484540228, −6.54901628821430004032311172176, −5.21147204726963166683233702870, −4.89892850274159833407510705487, −3.12284101882720555449499487054, −2.04731665134523931866722633506, −0.15649717172150991007035575650, 1.81876262299425626992108586709, 2.88181792382334428435581839427, 4.06553587797126204129162504379, 4.74196730598372758272984817831, 6.10813005071394217586285696486, 7.10578872476106693305261152465, 7.975796968143142447168750005183, 8.885127791478456243387638400257, 9.698970111736579651936416675460, 10.11520722473471453221698438522

Graph of the ZZ-function along the critical line