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Properties

Label 2-161-7.2-c3-0-8
Degree 22
Conductor 161161
Sign 0.942+0.334i0.942 + 0.334i
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  + (−2.21 + 3.83i)2-s + (−4.64 − 8.03i)3-s + (−5.80 − 10.0i)4-s + (−8.69 + 15.0i)5-s + 41.0·6-s + (−7.29 + 17.0i)7-s + 15.9·8-s + (−29.5 + 51.2i)9-s + (−38.5 − 66.7i)10-s + (−5.80 − 10.0i)11-s + (−53.8 + 93.2i)12-s − 43.4·13-s + (−49.1 − 65.6i)14-s + 161.·15-s + (11.1 − 19.2i)16-s + (27.5 + 47.7i)17-s + ⋯
L(s)  = 1  + (−0.782 + 1.35i)2-s + (−0.893 − 1.54i)3-s + (−0.725 − 1.25i)4-s + (−0.778 + 1.34i)5-s + 2.79·6-s + (−0.393 + 0.919i)7-s + 0.704·8-s + (−1.09 + 1.89i)9-s + (−1.21 − 2.10i)10-s + (−0.158 − 0.275i)11-s + (−1.29 + 2.24i)12-s − 0.926·13-s + (−0.938 − 1.25i)14-s + 2.78·15-s + (0.173 − 0.300i)16-s + (0.393 + 0.681i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((0.942+0.334i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+3/2) L(s)=((0.942+0.334i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: 0.942+0.334i0.942 + 0.334i
Analytic conductor: 9.499309.49930
Root analytic conductor: 3.082093.08209
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ161(93,⋅)\chi_{161} (93, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 161, ( :3/2), 0.942+0.334i)(2,\ 161,\ (\ :3/2),\ 0.942 + 0.334i)

Particular Values

L(2)L(2) ≈\approx 0.223200−0.0384518i0.223200 - 0.0384518i
L(12)L(\frac12) ≈\approx 0.223200−0.0384518i0.223200 - 0.0384518i
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+(7.29−17.0i)T 1 + (7.29 - 17.0i)T
23 1+(−11.5+19.9i)T 1 + (-11.5 + 19.9i)T
good2 1+(2.21−3.83i)T+(−4−6.92i)T2 1 + (2.21 - 3.83i)T + (-4 - 6.92i)T^{2}
3 1+(4.64+8.03i)T+(−13.5+23.3i)T2 1 + (4.64 + 8.03i)T + (-13.5 + 23.3i)T^{2}
5 1+(8.69−15.0i)T+(−62.5−108.i)T2 1 + (8.69 - 15.0i)T + (-62.5 - 108. i)T^{2}
11 1+(5.80+10.0i)T+(−665.5+1.15e3i)T2 1 + (5.80 + 10.0i)T + (-665.5 + 1.15e3i)T^{2}
13 1+43.4T+2.19e3T2 1 + 43.4T + 2.19e3T^{2}
17 1+(−27.5−47.7i)T+(−2.45e3+4.25e3i)T2 1 + (-27.5 - 47.7i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(−40.3+69.8i)T+(−3.42e3−5.94e3i)T2 1 + (-40.3 + 69.8i)T + (-3.42e3 - 5.94e3i)T^{2}
29 1+55.6T+2.43e4T2 1 + 55.6T + 2.43e4T^{2}
31 1+(120.+208.i)T+(−1.48e4+2.57e4i)T2 1 + (120. + 208. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(121.−211.i)T+(−2.53e4−4.38e4i)T2 1 + (121. - 211. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1−257.T+6.89e4T2 1 - 257.T + 6.89e4T^{2}
43 1−142.T+7.95e4T2 1 - 142.T + 7.95e4T^{2}
47 1+(−321.+556.i)T+(−5.19e4−8.99e4i)T2 1 + (-321. + 556. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(−241.−418.i)T+(−7.44e4+1.28e5i)T2 1 + (-241. - 418. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(−175.−303.i)T+(−1.02e5+1.77e5i)T2 1 + (-175. - 303. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(172.−298.i)T+(−1.13e5−1.96e5i)T2 1 + (172. - 298. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(−377.−654.i)T+(−1.50e5+2.60e5i)T2 1 + (-377. - 654. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+877.T+3.57e5T2 1 + 877.T + 3.57e5T^{2}
73 1+(612.+1.06e3i)T+(−1.94e5+3.36e5i)T2 1 + (612. + 1.06e3i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(110.−190.i)T+(−2.46e5−4.26e5i)T2 1 + (110. - 190. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1−416.T+5.71e5T2 1 - 416.T + 5.71e5T^{2}
89 1+(−770.+1.33e3i)T+(−3.52e5−6.10e5i)T2 1 + (-770. + 1.33e3i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1−754.T+9.12e5T2 1 - 754.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

−12.17150095083762564994086117471, −11.55339637411459652915213319512, −10.34449039277472381432251228107, −8.799656126933134184543522073944, −7.52973066828670495093712781734, −7.28455402369629245581666779020, −6.26910529537574978423923687827, −5.55899991185878296896945613507, −2.64732505038356379379760938663, −0.23526061101613678886231965893, 0.799186309028627392836823046450, 3.49010082805842236499957678395, 4.34872753086072239081410147626, 5.37340996747443438678375571896, 7.63246723690305675268177024860, 9.084188363252221092464312487064, 9.625191237087572855826264394883, 10.44224424145943689411889774344, 11.25781372732219325274850073468, 12.18582227174785526322738523460

Graph of the ZZ-function along the critical line