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Properties

Label 2-189-7.2-c3-0-23
Degree 22
Conductor 189189
Sign 0.843+0.537i0.843 + 0.537i
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  + (−1.74 + 3.02i)2-s + (−2.09 − 3.63i)4-s + (3.93 − 6.82i)5-s + (18.4 + 1.98i)7-s − 13.2·8-s + (13.7 + 23.8i)10-s + (−27.0 − 46.7i)11-s − 48.9·13-s + (−38.1 + 52.2i)14-s + (39.9 − 69.2i)16-s + (−48.4 − 83.9i)17-s + (71.1 − 123. i)19-s − 33.0·20-s + 188.·22-s + (−51.8 + 89.8i)23-s + ⋯
L(s)  = 1  + (−0.617 + 1.06i)2-s + (−0.262 − 0.454i)4-s + (0.352 − 0.610i)5-s + (0.994 + 0.107i)7-s − 0.587·8-s + (0.434 + 0.753i)10-s + (−0.740 − 1.28i)11-s − 1.04·13-s + (−0.728 + 0.996i)14-s + (0.624 − 1.08i)16-s + (−0.691 − 1.19i)17-s + (0.859 − 1.48i)19-s − 0.369·20-s + 1.82·22-s + (−0.470 + 0.814i)23-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=((0.843+0.537i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+3/2) L(s)=((0.843+0.537i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: 0.843+0.537i0.843 + 0.537i
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), 0.843+0.537i)(2,\ 189,\ (\ :3/2),\ 0.843 + 0.537i)

Particular Values

L(2)L(2) ≈\approx 0.917785−0.267728i0.917785 - 0.267728i
L(12)L(\frac12) ≈\approx 0.917785−0.267728i0.917785 - 0.267728i
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+(−18.4−1.98i)T 1 + (-18.4 - 1.98i)T
good2 1+(1.74−3.02i)T+(−4−6.92i)T2 1 + (1.74 - 3.02i)T + (-4 - 6.92i)T^{2}
5 1+(−3.93+6.82i)T+(−62.5−108.i)T2 1 + (-3.93 + 6.82i)T + (-62.5 - 108. i)T^{2}
11 1+(27.0+46.7i)T+(−665.5+1.15e3i)T2 1 + (27.0 + 46.7i)T + (-665.5 + 1.15e3i)T^{2}
13 1+48.9T+2.19e3T2 1 + 48.9T + 2.19e3T^{2}
17 1+(48.4+83.9i)T+(−2.45e3+4.25e3i)T2 1 + (48.4 + 83.9i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(−71.1+123.i)T+(−3.42e3−5.94e3i)T2 1 + (-71.1 + 123. i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(51.8−89.8i)T+(−6.08e3−1.05e4i)T2 1 + (51.8 - 89.8i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1−24.5T+2.43e4T2 1 - 24.5T + 2.43e4T^{2}
31 1+(93.6+162.i)T+(−1.48e4+2.57e4i)T2 1 + (93.6 + 162. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(73.2−126.i)T+(−2.53e4−4.38e4i)T2 1 + (73.2 - 126. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1−314.T+6.89e4T2 1 - 314.T + 6.89e4T^{2}
43 1−173.T+7.95e4T2 1 - 173.T + 7.95e4T^{2}
47 1+(−129.+224.i)T+(−5.19e4−8.99e4i)T2 1 + (-129. + 224. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(310.+537.i)T+(−7.44e4+1.28e5i)T2 1 + (310. + 537. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(−221.−383.i)T+(−1.02e5+1.77e5i)T2 1 + (-221. - 383. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(56.6−98.1i)T+(−1.13e5−1.96e5i)T2 1 + (56.6 - 98.1i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(314.+544.i)T+(−1.50e5+2.60e5i)T2 1 + (314. + 544. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1−41.3T+3.57e5T2 1 - 41.3T + 3.57e5T^{2}
73 1+(223.+387.i)T+(−1.94e5+3.36e5i)T2 1 + (223. + 387. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(217.−376.i)T+(−2.46e5−4.26e5i)T2 1 + (217. - 376. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+329.T+5.71e5T2 1 + 329.T + 5.71e5T^{2}
89 1+(−12.4+21.5i)T+(−3.52e5−6.10e5i)T2 1 + (-12.4 + 21.5i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+499.T+9.12e5T2 1 + 499.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

−11.84160198059153840577980385550, −11.12512942344965713802264789007, −9.501810281770336789332618451345, −8.908591262799585855045905838991, −7.86424267775512136303932295376, −7.14363505408832916263204057777, −5.60490276054673422627694834325, −4.97423309235453334017292077473, −2.69988384861738168456504506030, −0.50359374107528565397000989736, 1.69107666540644438352241199727, 2.56285346623447984618753019981, 4.35322549215037207751689119614, 5.80685679639848075038461250636, 7.29042880421455211539452902099, 8.302064542026628809184483041727, 9.609574435011028436742553354797, 10.41978712387078528048406123747, 10.83676558361308689762721642861, 12.22455516274871071659851517958

Graph of the ZZ-function along the critical line