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Properties

Label 2-161-7.2-c3-0-6
Degree 22
Conductor 161161
Sign 0.382+0.924i0.382 + 0.924i
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  + (−2.44 + 4.23i)2-s + (3.42 + 5.92i)3-s + (−7.98 − 13.8i)4-s + (−9.57 + 16.5i)5-s − 33.4·6-s + (−17.5 + 5.98i)7-s + 38.9·8-s + (−9.89 + 17.1i)9-s + (−46.8 − 81.1i)10-s + (27.4 + 47.5i)11-s + (54.5 − 94.5i)12-s + 53.1·13-s + (17.5 − 88.9i)14-s − 130.·15-s + (−31.5 + 54.5i)16-s + (−13.7 − 23.7i)17-s + ⋯
L(s)  = 1  + (−0.865 + 1.49i)2-s + (0.658 + 1.14i)3-s + (−0.997 − 1.72i)4-s + (−0.856 + 1.48i)5-s − 2.27·6-s + (−0.946 + 0.322i)7-s + 1.72·8-s + (−0.366 + 0.634i)9-s + (−1.48 − 2.56i)10-s + (0.751 + 1.30i)11-s + (1.31 − 2.27i)12-s + 1.13·13-s + (0.334 − 1.69i)14-s − 2.25·15-s + (−0.492 + 0.853i)16-s + (−0.195 − 0.339i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((0.382+0.924i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+3/2) L(s)=((0.382+0.924i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: 0.382+0.924i0.382 + 0.924i
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), 0.382+0.924i)(2,\ 161,\ (\ :3/2),\ 0.382 + 0.924i)

Particular Values

L(2)L(2) ≈\approx 0.687489−0.459614i0.687489 - 0.459614i
L(12)L(\frac12) ≈\approx 0.687489−0.459614i0.687489 - 0.459614i
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+(17.5−5.98i)T 1 + (17.5 - 5.98i)T
23 1+(−11.5+19.9i)T 1 + (-11.5 + 19.9i)T
good2 1+(2.44−4.23i)T+(−4−6.92i)T2 1 + (2.44 - 4.23i)T + (-4 - 6.92i)T^{2}
3 1+(−3.42−5.92i)T+(−13.5+23.3i)T2 1 + (-3.42 - 5.92i)T + (-13.5 + 23.3i)T^{2}
5 1+(9.57−16.5i)T+(−62.5−108.i)T2 1 + (9.57 - 16.5i)T + (-62.5 - 108. i)T^{2}
11 1+(−27.4−47.5i)T+(−665.5+1.15e3i)T2 1 + (-27.4 - 47.5i)T + (-665.5 + 1.15e3i)T^{2}
13 1−53.1T+2.19e3T2 1 - 53.1T + 2.19e3T^{2}
17 1+(13.7+23.7i)T+(−2.45e3+4.25e3i)T2 1 + (13.7 + 23.7i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(43.6−75.6i)T+(−3.42e3−5.94e3i)T2 1 + (43.6 - 75.6i)T + (-3.42e3 - 5.94e3i)T^{2}
29 1−285.T+2.43e4T2 1 - 285.T + 2.43e4T^{2}
31 1+(−28.1−48.7i)T+(−1.48e4+2.57e4i)T2 1 + (-28.1 - 48.7i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(8.63−14.9i)T+(−2.53e4−4.38e4i)T2 1 + (8.63 - 14.9i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1+11.3T+6.89e4T2 1 + 11.3T + 6.89e4T^{2}
43 1+155.T+7.95e4T2 1 + 155.T + 7.95e4T^{2}
47 1+(67.9−117.i)T+(−5.19e4−8.99e4i)T2 1 + (67.9 - 117. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(150.+261.i)T+(−7.44e4+1.28e5i)T2 1 + (150. + 261. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(−85.7−148.i)T+(−1.02e5+1.77e5i)T2 1 + (-85.7 - 148. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(245.−424.i)T+(−1.13e5−1.96e5i)T2 1 + (245. - 424. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(510.+884.i)T+(−1.50e5+2.60e5i)T2 1 + (510. + 884. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+32.0T+3.57e5T2 1 + 32.0T + 3.57e5T^{2}
73 1+(−359.−622.i)T+(−1.94e5+3.36e5i)T2 1 + (-359. - 622. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(−549.+952.i)T+(−2.46e5−4.26e5i)T2 1 + (-549. + 952. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+433.T+5.71e5T2 1 + 433.T + 5.71e5T^{2}
89 1+(3.26−5.65i)T+(−3.52e5−6.10e5i)T2 1 + (3.26 - 5.65i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1−700.T+9.12e5T2 1 - 700.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

−13.97646229583503327351944255517, −12.08768564262383500078905660378, −10.52129723771774873578026710410, −9.972874355222017008513426278837, −9.036291772860080062129189924180, −8.111572690322306569009070161900, −6.85210944572084519545884125895, −6.35850506033762738252124567631, −4.35963008619190097644236972209, −3.20879462030168729458170596729, 0.56391831399791762728735765070, 1.26764670287572808557546373062, 3.07327970147029831147329531047, 4.10589545057354926667568582039, 6.51590746299780301302324221933, 8.131790402057496220964752440258, 8.589707733135364975206433319248, 9.226566173866518862725235134617, 10.76476509944998198110456651233, 11.74613719181295690391649728970

Graph of the ZZ-function along the critical line