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Properties

Label 2-161-7.4-c3-0-33
Degree 22
Conductor 161161
Sign 0.985+0.169i0.985 + 0.169i
Analytic cond. 9.499309.49930
Root an. cond. 3.082093.08209
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  + (2.08 + 3.61i)2-s + (3.80 − 6.59i)3-s + (−4.70 + 8.15i)4-s + (−1.49 − 2.59i)5-s + 31.8·6-s + (1.97 − 18.4i)7-s − 5.91·8-s + (−15.5 − 26.9i)9-s + (6.24 − 10.8i)10-s + (27.0 − 46.7i)11-s + (35.8 + 62.1i)12-s − 50.0·13-s + (70.6 − 31.2i)14-s − 22.7·15-s + (25.3 + 43.8i)16-s + (−16.4 + 28.4i)17-s + ⋯
L(s)  = 1  + (0.737 + 1.27i)2-s + (0.733 − 1.26i)3-s + (−0.588 + 1.01i)4-s + (−0.133 − 0.231i)5-s + 2.16·6-s + (0.106 − 0.994i)7-s − 0.261·8-s + (−0.575 − 0.996i)9-s + (0.197 − 0.341i)10-s + (0.740 − 1.28i)11-s + (0.863 + 1.49i)12-s − 1.06·13-s + (1.34 − 0.597i)14-s − 0.392·15-s + (0.395 + 0.685i)16-s + (−0.234 + 0.406i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((0.985+0.169i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+3/2) L(s)=((0.985+0.169i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: 0.985+0.169i0.985 + 0.169i
Analytic conductor: 9.499309.49930
Root analytic conductor: 3.082093.08209
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ161(116,⋅)\chi_{161} (116, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 161, ( :3/2), 0.985+0.169i)(2,\ 161,\ (\ :3/2),\ 0.985 + 0.169i)

Particular Values

L(2)L(2) ≈\approx 3.06510−0.261821i3.06510 - 0.261821i
L(12)L(\frac12) ≈\approx 3.06510−0.261821i3.06510 - 0.261821i
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)
bad7 1+(−1.97+18.4i)T 1 + (-1.97 + 18.4i)T
23 1+(−11.5−19.9i)T 1 + (-11.5 - 19.9i)T
good2 1+(−2.08−3.61i)T+(−4+6.92i)T2 1 + (-2.08 - 3.61i)T + (-4 + 6.92i)T^{2}
3 1+(−3.80+6.59i)T+(−13.5−23.3i)T2 1 + (-3.80 + 6.59i)T + (-13.5 - 23.3i)T^{2}
5 1+(1.49+2.59i)T+(−62.5+108.i)T2 1 + (1.49 + 2.59i)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+50.0T+2.19e3T2 1 + 50.0T + 2.19e3T^{2}
17 1+(16.4−28.4i)T+(−2.45e3−4.25e3i)T2 1 + (16.4 - 28.4i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(−40.6−70.4i)T+(−3.42e3+5.94e3i)T2 1 + (-40.6 - 70.4i)T + (-3.42e3 + 5.94e3i)T^{2}
29 1−244.T+2.43e4T2 1 - 244.T + 2.43e4T^{2}
31 1+(90.7−157.i)T+(−1.48e4−2.57e4i)T2 1 + (90.7 - 157. i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(−131.−227.i)T+(−2.53e4+4.38e4i)T2 1 + (-131. - 227. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1+442.T+6.89e4T2 1 + 442.T + 6.89e4T^{2}
43 1+194.T+7.95e4T2 1 + 194.T + 7.95e4T^{2}
47 1+(−167.−289.i)T+(−5.19e4+8.99e4i)T2 1 + (-167. - 289. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(−139.+240.i)T+(−7.44e4−1.28e5i)T2 1 + (-139. + 240. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(43.8−75.9i)T+(−1.02e5−1.77e5i)T2 1 + (43.8 - 75.9i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(97.4+168.i)T+(−1.13e5+1.96e5i)T2 1 + (97.4 + 168. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(−304.+526.i)T+(−1.50e5−2.60e5i)T2 1 + (-304. + 526. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+890.T+3.57e5T2 1 + 890.T + 3.57e5T^{2}
73 1+(250.−433.i)T+(−1.94e5−3.36e5i)T2 1 + (250. - 433. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(−323.−560.i)T+(−2.46e5+4.26e5i)T2 1 + (-323. - 560. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1−94.1T+5.71e5T2 1 - 94.1T + 5.71e5T^{2}
89 1+(−325.−563.i)T+(−3.52e5+6.10e5i)T2 1 + (-325. - 563. i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−968.T+9.12e5T2 1 - 968.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

−12.81740959460830774355541995192, −11.90826085016943756390164926448, −10.32202127510183059701123287696, −8.545710490812626988200113512678, −7.972180031900628505161106505258, −6.97533280217612585389058364310, −6.33861564022398046587583935177, −4.78891836134948334685263736394, −3.37197412523759804357971265840, −1.21995430252630831343713051233, 2.23082636346589063586961494726, 3.13522292194602744834523954897, 4.43794807541020757959538469419, 5.06025595759102783140474927124, 7.21226682674082754145299992873, 8.944622023780925961520543642654, 9.632543087765998692411576436615, 10.39240452562517051669601557708, 11.62889525425318494011236629271, 12.16479081425769103923966825800

Graph of the ZZ-function along the critical line