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Properties

Label 2-161-7.4-c3-0-4
Degree 22
Conductor 161161
Sign 0.279+0.960i0.279 + 0.960i
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  + (2.42 + 4.19i)2-s + (−4.52 + 7.83i)3-s + (−7.73 + 13.3i)4-s + (−7.08 − 12.2i)5-s − 43.8·6-s + (17.4 − 6.30i)7-s − 36.1·8-s + (−27.4 − 47.5i)9-s + (34.2 − 59.4i)10-s + (−24.1 + 41.8i)11-s + (−69.9 − 121. i)12-s − 21.7·13-s + (68.6 + 57.7i)14-s + 128.·15-s + (−25.6 − 44.4i)16-s + (−54.5 + 94.5i)17-s + ⋯
L(s)  = 1  + (0.856 + 1.48i)2-s + (−0.870 + 1.50i)3-s + (−0.966 + 1.67i)4-s + (−0.633 − 1.09i)5-s − 2.98·6-s + (0.940 − 0.340i)7-s − 1.59·8-s + (−1.01 − 1.75i)9-s + (1.08 − 1.87i)10-s + (−0.661 + 1.14i)11-s + (−1.68 − 2.91i)12-s − 0.464·13-s + (1.30 + 1.10i)14-s + 2.20·15-s + (−0.401 − 0.694i)16-s + (−0.778 + 1.34i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((0.279+0.960i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+3/2) L(s)=((0.279+0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: 0.279+0.960i0.279 + 0.960i
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), 0.279+0.960i)(2,\ 161,\ (\ :3/2),\ 0.279 + 0.960i)

Particular Values

L(2)L(2) ≈\approx 0.653339−0.490026i0.653339 - 0.490026i
L(12)L(\frac12) ≈\approx 0.653339−0.490026i0.653339 - 0.490026i
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+(−17.4+6.30i)T 1 + (-17.4 + 6.30i)T
23 1+(−11.5−19.9i)T 1 + (-11.5 - 19.9i)T
good2 1+(−2.42−4.19i)T+(−4+6.92i)T2 1 + (-2.42 - 4.19i)T + (-4 + 6.92i)T^{2}
3 1+(4.52−7.83i)T+(−13.5−23.3i)T2 1 + (4.52 - 7.83i)T + (-13.5 - 23.3i)T^{2}
5 1+(7.08+12.2i)T+(−62.5+108.i)T2 1 + (7.08 + 12.2i)T + (-62.5 + 108. i)T^{2}
11 1+(24.1−41.8i)T+(−665.5−1.15e3i)T2 1 + (24.1 - 41.8i)T + (-665.5 - 1.15e3i)T^{2}
13 1+21.7T+2.19e3T2 1 + 21.7T + 2.19e3T^{2}
17 1+(54.5−94.5i)T+(−2.45e3−4.25e3i)T2 1 + (54.5 - 94.5i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(59.9+103.i)T+(−3.42e3+5.94e3i)T2 1 + (59.9 + 103. i)T + (-3.42e3 + 5.94e3i)T^{2}
29 1−85.5T+2.43e4T2 1 - 85.5T + 2.43e4T^{2}
31 1+(−36.8+63.8i)T+(−1.48e4−2.57e4i)T2 1 + (-36.8 + 63.8i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(−127.−220.i)T+(−2.53e4+4.38e4i)T2 1 + (-127. - 220. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1+108.T+6.89e4T2 1 + 108.T + 6.89e4T^{2}
43 1+447.T+7.95e4T2 1 + 447.T + 7.95e4T^{2}
47 1+(−25.3−43.8i)T+(−5.19e4+8.99e4i)T2 1 + (-25.3 - 43.8i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(310.−538.i)T+(−7.44e4−1.28e5i)T2 1 + (310. - 538. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(−234.+406.i)T+(−1.02e5−1.77e5i)T2 1 + (-234. + 406. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(87.0+150.i)T+(−1.13e5+1.96e5i)T2 1 + (87.0 + 150. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(138.−239.i)T+(−1.50e5−2.60e5i)T2 1 + (138. - 239. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1−77.7T+3.57e5T2 1 - 77.7T + 3.57e5T^{2}
73 1+(433.−750.i)T+(−1.94e5−3.36e5i)T2 1 + (433. - 750. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(−197.−342.i)T+(−2.46e5+4.26e5i)T2 1 + (-197. - 342. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+207.T+5.71e5T2 1 + 207.T + 5.71e5T^{2}
89 1+(−494.−856.i)T+(−3.52e5+6.10e5i)T2 1 + (-494. - 856. i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1+1.13e3T+9.12e5T2 1 + 1.13e3T + 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

−13.24387976961324709628834258082, −12.39392607398430717951566130140, −11.31363901566550549101054800499, −10.20645367643904643034023959383, −8.798235302636033032717058771793, −7.971033605132359105381519440996, −6.58622772076960866956139131054, −5.15064927672222944332523774315, −4.64909306679849075586810683922, −4.19652640848887627645793995794, 0.31578639625973095296821880056, 1.96147358028849628302844177379, 3.01491626670708281082775994472, 4.85555270286815958624898500393, 5.94352884051044092382264916473, 7.20262312604261990252893198156, 8.288087698340973419873347331220, 10.43802169213185799753571741648, 11.18187515503977593521293637386, 11.61006625232740220536765120100

Graph of the ZZ-function along the critical line