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Properties

Label 2-189-189.101-c1-0-5
Degree 22
Conductor 189189
Sign −0.213−0.976i-0.213 - 0.976i
Analytic cond. 1.509171.50917
Root an. cond. 1.228481.22848
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  + (−1.60 + 1.91i)2-s + (−1.44 − 0.952i)3-s + (−0.733 − 4.16i)4-s + (−0.273 + 0.229i)5-s + (4.14 − 1.23i)6-s + (2.61 − 0.393i)7-s + (4.81 + 2.77i)8-s + (1.18 + 2.75i)9-s − 0.892i·10-s + (−2.21 + 2.63i)11-s + (−2.90 + 6.71i)12-s + (−0.362 + 0.995i)13-s + (−3.44 + 5.63i)14-s + (0.615 − 0.0714i)15-s + (−5.08 + 1.85i)16-s + 5.82·17-s + ⋯
L(s)  = 1  + (−1.13 + 1.35i)2-s + (−0.835 − 0.550i)3-s + (−0.366 − 2.08i)4-s + (−0.122 + 0.102i)5-s + (1.69 − 0.504i)6-s + (0.988 − 0.148i)7-s + (1.70 + 0.982i)8-s + (0.394 + 0.918i)9-s − 0.282i·10-s + (−0.667 + 0.795i)11-s + (−0.838 + 1.93i)12-s + (−0.100 + 0.276i)13-s + (−0.920 + 1.50i)14-s + (0.158 − 0.0184i)15-s + (−1.27 + 0.462i)16-s + 1.41·17-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=((−0.213−0.976i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+1/2) L(s)=((−0.213−0.976i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: −0.213−0.976i-0.213 - 0.976i
Analytic conductor: 1.509171.50917
Root analytic conductor: 1.228481.22848
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ189(101,⋅)\chi_{189} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 189, ( :1/2), −0.213−0.976i)(2,\ 189,\ (\ :1/2),\ -0.213 - 0.976i)

Particular Values

L(1)L(1) ≈\approx 0.318734+0.396084i0.318734 + 0.396084i
L(12)L(\frac12) ≈\approx 0.318734+0.396084i0.318734 + 0.396084i
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)
bad3 1+(1.44+0.952i)T 1 + (1.44 + 0.952i)T
7 1+(−2.61+0.393i)T 1 + (-2.61 + 0.393i)T
good2 1+(1.60−1.91i)T+(−0.347−1.96i)T2 1 + (1.60 - 1.91i)T + (-0.347 - 1.96i)T^{2}
5 1+(0.273−0.229i)T+(0.868−4.92i)T2 1 + (0.273 - 0.229i)T + (0.868 - 4.92i)T^{2}
11 1+(2.21−2.63i)T+(−1.91−10.8i)T2 1 + (2.21 - 2.63i)T + (-1.91 - 10.8i)T^{2}
13 1+(0.362−0.995i)T+(−9.95−8.35i)T2 1 + (0.362 - 0.995i)T + (-9.95 - 8.35i)T^{2}
17 1−5.82T+17T2 1 - 5.82T + 17T^{2}
19 1−4.08iT−19T2 1 - 4.08iT - 19T^{2}
23 1+(−0.737+2.02i)T+(−17.6−14.7i)T2 1 + (-0.737 + 2.02i)T + (-17.6 - 14.7i)T^{2}
29 1+(−2.91−7.99i)T+(−22.2+18.6i)T2 1 + (-2.91 - 7.99i)T + (-22.2 + 18.6i)T^{2}
31 1+(−3.71+0.655i)T+(29.1−10.6i)T2 1 + (-3.71 + 0.655i)T + (29.1 - 10.6i)T^{2}
37 1+(−0.937+1.62i)T+(−18.5−32.0i)T2 1 + (-0.937 + 1.62i)T + (-18.5 - 32.0i)T^{2}
41 1+(−4.24−1.54i)T+(31.4+26.3i)T2 1 + (-4.24 - 1.54i)T + (31.4 + 26.3i)T^{2}
43 1+(1.24−7.08i)T+(−40.4−14.7i)T2 1 + (1.24 - 7.08i)T + (-40.4 - 14.7i)T^{2}
47 1+(−2.12+12.0i)T+(−44.1−16.0i)T2 1 + (-2.12 + 12.0i)T + (-44.1 - 16.0i)T^{2}
53 1+(7.91+4.56i)T+(26.5+45.8i)T2 1 + (7.91 + 4.56i)T + (26.5 + 45.8i)T^{2}
59 1+(3.87+1.41i)T+(45.1+37.9i)T2 1 + (3.87 + 1.41i)T + (45.1 + 37.9i)T^{2}
61 1+(−13.0−2.30i)T+(57.3+20.8i)T2 1 + (-13.0 - 2.30i)T + (57.3 + 20.8i)T^{2}
67 1+(−3.53+2.96i)T+(11.6−65.9i)T2 1 + (-3.53 + 2.96i)T + (11.6 - 65.9i)T^{2}
71 1+(7.24−4.18i)T+(35.5−61.4i)T2 1 + (7.24 - 4.18i)T + (35.5 - 61.4i)T^{2}
73 1+(6.45−3.72i)T+(36.5−63.2i)T2 1 + (6.45 - 3.72i)T + (36.5 - 63.2i)T^{2}
79 1+(10.2+8.62i)T+(13.7+77.7i)T2 1 + (10.2 + 8.62i)T + (13.7 + 77.7i)T^{2}
83 1+(−3.80+1.38i)T+(63.5−53.3i)T2 1 + (-3.80 + 1.38i)T + (63.5 - 53.3i)T^{2}
89 1+11.5T+89T2 1 + 11.5T + 89T^{2}
97 1+(−4.45−0.785i)T+(91.1+33.1i)T2 1 + (-4.45 - 0.785i)T + (91.1 + 33.1i)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

−12.79590332530387955443219980764, −11.71519659021429470056293493448, −10.55733226859129629475440781804, −9.886660634107762554339706745118, −8.360280200150922426349765006666, −7.64547765769815080801895968325, −6.97815347251006639836427368567, −5.69580929056151017250369043353, −4.90256895098033179794704415660, −1.41978014533203601293627758965, 0.852340565517255519102660794241, 2.86482334881823590306374278603, 4.38941288305934384581787914350, 5.71803383026809459111761843965, 7.69388069299922639133595511613, 8.492204822837719763515926361997, 9.660194079163534824013749499160, 10.43633565350735687243658102009, 11.20989146672668147925600962680, 11.82742856595989492301584505654

Graph of the ZZ-function along the critical line