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Properties

Label 2-189-7.2-c3-0-6
Degree 22
Conductor 189189
Sign −0.390+0.920i-0.390 + 0.920i
Analytic cond. 11.151311.1513
Root an. cond. 3.339363.33936
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.64 + 2.85i)2-s + (−1.42 − 2.46i)4-s + (−10.8 + 18.8i)5-s + (1.08 + 18.4i)7-s − 16.9·8-s + (−35.7 − 61.9i)10-s + (20.9 + 36.3i)11-s + 46.0·13-s + (−54.5 − 27.3i)14-s + (39.3 − 68.1i)16-s + (−1.00 − 1.74i)17-s + (−36.6 + 63.4i)19-s + 61.7·20-s − 138.·22-s + (12.0 − 20.9i)23-s + ⋯
L(s)  = 1  + (−0.582 + 1.00i)2-s + (−0.177 − 0.307i)4-s + (−0.971 + 1.68i)5-s + (0.0587 + 0.998i)7-s − 0.750·8-s + (−1.13 − 1.95i)10-s + (0.575 + 0.996i)11-s + 0.982·13-s + (−1.04 − 0.521i)14-s + (0.614 − 1.06i)16-s + (−0.0143 − 0.0249i)17-s + (−0.442 + 0.765i)19-s + 0.690·20-s − 1.33·22-s + (0.109 − 0.189i)23-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=((−0.390+0.920i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+3/2) L(s)=((−0.390+0.920i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: −0.390+0.920i-0.390 + 0.920i
Analytic conductor: 11.151311.1513
Root analytic conductor: 3.339363.33936
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ189(163,⋅)\chi_{189} (163, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 189, ( :3/2), −0.390+0.920i)(2,\ 189,\ (\ :3/2),\ -0.390 + 0.920i)

Particular Values

L(2)L(2) ≈\approx 0.473036−0.714706i0.473036 - 0.714706i
L(12)L(\frac12) ≈\approx 0.473036−0.714706i0.473036 - 0.714706i
L(52)L(\frac{5}{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
7 1+(−1.08−18.4i)T 1 + (-1.08 - 18.4i)T
good2 1+(1.64−2.85i)T+(−4−6.92i)T2 1 + (1.64 - 2.85i)T + (-4 - 6.92i)T^{2}
5 1+(10.8−18.8i)T+(−62.5−108.i)T2 1 + (10.8 - 18.8i)T + (-62.5 - 108. i)T^{2}
11 1+(−20.9−36.3i)T+(−665.5+1.15e3i)T2 1 + (-20.9 - 36.3i)T + (-665.5 + 1.15e3i)T^{2}
13 1−46.0T+2.19e3T2 1 - 46.0T + 2.19e3T^{2}
17 1+(1.00+1.74i)T+(−2.45e3+4.25e3i)T2 1 + (1.00 + 1.74i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(36.6−63.4i)T+(−3.42e3−5.94e3i)T2 1 + (36.6 - 63.4i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(−12.0+20.9i)T+(−6.08e3−1.05e4i)T2 1 + (-12.0 + 20.9i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1−90.7T+2.43e4T2 1 - 90.7T + 2.43e4T^{2}
31 1+(26.6+46.0i)T+(−1.48e4+2.57e4i)T2 1 + (26.6 + 46.0i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(33.9−58.7i)T+(−2.53e4−4.38e4i)T2 1 + (33.9 - 58.7i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1−341.T+6.89e4T2 1 - 341.T + 6.89e4T^{2}
43 1−509.T+7.95e4T2 1 - 509.T + 7.95e4T^{2}
47 1+(19.2−33.3i)T+(−5.19e4−8.99e4i)T2 1 + (19.2 - 33.3i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(194.+337.i)T+(−7.44e4+1.28e5i)T2 1 + (194. + 337. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(225.+390.i)T+(−1.02e5+1.77e5i)T2 1 + (225. + 390. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(112.−195.i)T+(−1.13e5−1.96e5i)T2 1 + (112. - 195. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(−386.−668.i)T+(−1.50e5+2.60e5i)T2 1 + (-386. - 668. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+962.T+3.57e5T2 1 + 962.T + 3.57e5T^{2}
73 1+(−526.−912.i)T+(−1.94e5+3.36e5i)T2 1 + (-526. - 912. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(16.7−29.0i)T+(−2.46e5−4.26e5i)T2 1 + (16.7 - 29.0i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1−446.T+5.71e5T2 1 - 446.T + 5.71e5T^{2}
89 1+(244.−424.i)T+(−3.52e5−6.10e5i)T2 1 + (244. - 424. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1−460.T+9.12e5T2 1 - 460.T + 9.12e5T^{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

−12.50156371498003426183814060290, −11.74118144847326328156664238712, −10.83670055441241586002439381129, −9.602070183351491963083685606801, −8.447638826174167158924358938957, −7.63394452691974640066024732026, −6.69510037953115278636857381763, −6.01707208465473661608137347334, −3.92897868005878531710629513532, −2.62565952314023300130763645709, 0.53717759926566335932561418132, 1.22421567805574216186788462594, 3.51180788564657472370923776728, 4.46007006849155139650222886666, 6.04672004892247760558777528801, 7.73791995455149475669257034367, 8.779424817691257467898941700150, 9.208407024993750800777556320342, 10.77825016796798550169995450953, 11.25898807366775336867792372209

Graph of the ZZ-function along the critical line