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Properties

Label 2-161-7.4-c3-0-12
Degree 22
Conductor 161161
Sign −0.998+0.0620i-0.998 + 0.0620i
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  + (1.15 + 2.00i)2-s + (−4.37 + 7.57i)3-s + (1.32 − 2.28i)4-s + (7.70 + 13.3i)5-s − 20.2·6-s + (2.31 + 18.3i)7-s + 24.6·8-s + (−24.7 − 42.8i)9-s + (−17.8 + 30.9i)10-s + (−24.3 + 42.2i)11-s + (11.5 + 19.9i)12-s + 53.5·13-s + (−34.1 + 25.9i)14-s − 134.·15-s + (17.9 + 31.0i)16-s + (42.8 − 74.3i)17-s + ⋯
L(s)  = 1  + (0.409 + 0.708i)2-s + (−0.841 + 1.45i)3-s + (0.165 − 0.285i)4-s + (0.689 + 1.19i)5-s − 1.37·6-s + (0.125 + 0.992i)7-s + 1.08·8-s + (−0.915 − 1.58i)9-s + (−0.564 + 0.977i)10-s + (−0.668 + 1.15i)11-s + (0.277 + 0.480i)12-s + 1.14·13-s + (−0.652 + 0.494i)14-s − 2.32·15-s + (0.280 + 0.485i)16-s + (0.612 − 1.06i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((−0.998+0.0620i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0620i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+3/2) L(s)=((−0.998+0.0620i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0620i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: −0.998+0.0620i-0.998 + 0.0620i
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), −0.998+0.0620i)(2,\ 161,\ (\ :3/2),\ -0.998 + 0.0620i)

Particular Values

L(2)L(2) ≈\approx 0.0594771−1.91498i0.0594771 - 1.91498i
L(12)L(\frac12) ≈\approx 0.0594771−1.91498i0.0594771 - 1.91498i
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+(−2.31−18.3i)T 1 + (-2.31 - 18.3i)T
23 1+(−11.5−19.9i)T 1 + (-11.5 - 19.9i)T
good2 1+(−1.15−2.00i)T+(−4+6.92i)T2 1 + (-1.15 - 2.00i)T + (-4 + 6.92i)T^{2}
3 1+(4.37−7.57i)T+(−13.5−23.3i)T2 1 + (4.37 - 7.57i)T + (-13.5 - 23.3i)T^{2}
5 1+(−7.70−13.3i)T+(−62.5+108.i)T2 1 + (-7.70 - 13.3i)T + (-62.5 + 108. i)T^{2}
11 1+(24.3−42.2i)T+(−665.5−1.15e3i)T2 1 + (24.3 - 42.2i)T + (-665.5 - 1.15e3i)T^{2}
13 1−53.5T+2.19e3T2 1 - 53.5T + 2.19e3T^{2}
17 1+(−42.8+74.3i)T+(−2.45e3−4.25e3i)T2 1 + (-42.8 + 74.3i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(64.1+111.i)T+(−3.42e3+5.94e3i)T2 1 + (64.1 + 111. i)T + (-3.42e3 + 5.94e3i)T^{2}
29 1+133.T+2.43e4T2 1 + 133.T + 2.43e4T^{2}
31 1+(−41.4+71.8i)T+(−1.48e4−2.57e4i)T2 1 + (-41.4 + 71.8i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(27.1+47.0i)T+(−2.53e4+4.38e4i)T2 1 + (27.1 + 47.0i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1−475.T+6.89e4T2 1 - 475.T + 6.89e4T^{2}
43 1+146.T+7.95e4T2 1 + 146.T + 7.95e4T^{2}
47 1+(−261.−453.i)T+(−5.19e4+8.99e4i)T2 1 + (-261. - 453. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(−135.+235.i)T+(−7.44e4−1.28e5i)T2 1 + (-135. + 235. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(87.0−150.i)T+(−1.02e5−1.77e5i)T2 1 + (87.0 - 150. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(−225.−389.i)T+(−1.13e5+1.96e5i)T2 1 + (-225. - 389. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(297.−515.i)T+(−1.50e5−2.60e5i)T2 1 + (297. - 515. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1−438.T+3.57e5T2 1 - 438.T + 3.57e5T^{2}
73 1+(300.−520.i)T+(−1.94e5−3.36e5i)T2 1 + (300. - 520. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(435.+755.i)T+(−2.46e5+4.26e5i)T2 1 + (435. + 755. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1−229.T+5.71e5T2 1 - 229.T + 5.71e5T^{2}
89 1+(501.+868.i)T+(−3.52e5+6.10e5i)T2 1 + (501. + 868. i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−1.55e3T+9.12e5T2 1 - 1.55e3T + 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

−13.11023959510334433672216845043, −11.45854117721217493278345698703, −10.86438163870728398642358072515, −10.05958089755585207204374564864, −9.223933552137216810980496440032, −7.24320520928646031003741915670, −6.11106795951858242435189952912, −5.48939597669704843438287145723, −4.46829978885398160739982867814, −2.57819913427835631728047272779, 0.940729231782780909445014866902, 1.75305631700973701653147815684, 3.81823586870699581931226354216, 5.46459343831939300121000614412, 6.30723993109545275175380568943, 7.79101405133463191581034000068, 8.401771954803520521069399783689, 10.49909816989197243721550633467, 11.04589278159378544614560295446, 12.22199167858069696057840394272

Graph of the ZZ-function along the critical line