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Properties

Label 2-161-161.10-c1-0-7
Degree 22
Conductor 161161
Sign 0.866+0.499i0.866 + 0.499i
Analytic cond. 1.285591.28559
Root an. cond. 1.133831.13383
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.318 − 0.919i)2-s + (0.100 + 0.194i)3-s + (0.827 + 0.650i)4-s + (−1.63 + 0.156i)5-s + (0.210 − 0.0303i)6-s + (2.63 + 0.196i)7-s + (2.49 − 1.60i)8-s + (1.71 − 2.40i)9-s + (−0.376 + 1.55i)10-s + (−1.66 + 0.576i)11-s + (−0.0436 + 0.226i)12-s + (0.504 + 1.71i)13-s + (1.02 − 2.36i)14-s + (−0.194 − 0.302i)15-s + (−0.185 − 0.765i)16-s + (−1.46 − 0.588i)17-s + ⋯
L(s)  = 1  + (0.225 − 0.650i)2-s + (0.0579 + 0.112i)3-s + (0.413 + 0.325i)4-s + (−0.731 + 0.0698i)5-s + (0.0861 − 0.0123i)6-s + (0.997 + 0.0743i)7-s + (0.883 − 0.567i)8-s + (0.570 − 0.801i)9-s + (−0.119 + 0.491i)10-s + (−0.501 + 0.173i)11-s + (−0.0125 + 0.0653i)12-s + (0.139 + 0.476i)13-s + (0.272 − 0.631i)14-s + (−0.0501 − 0.0781i)15-s + (−0.0464 − 0.191i)16-s + (−0.356 − 0.142i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((0.866+0.499i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+1/2) L(s)=((0.866+0.499i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: 0.866+0.499i0.866 + 0.499i
Analytic conductor: 1.285591.28559
Root analytic conductor: 1.133831.13383
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ161(10,⋅)\chi_{161} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 161, ( :1/2), 0.866+0.499i)(2,\ 161,\ (\ :1/2),\ 0.866 + 0.499i)

Particular Values

L(1)L(1) ≈\approx 1.37085−0.366550i1.37085 - 0.366550i
L(12)L(\frac12) ≈\approx 1.37085−0.366550i1.37085 - 0.366550i
L(32)L(\frac{3}{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+(−2.63−0.196i)T 1 + (-2.63 - 0.196i)T
23 1+(4.30+2.11i)T 1 + (4.30 + 2.11i)T
good2 1+(−0.318+0.919i)T+(−1.57−1.23i)T2 1 + (-0.318 + 0.919i)T + (-1.57 - 1.23i)T^{2}
3 1+(−0.100−0.194i)T+(−1.74+2.44i)T2 1 + (-0.100 - 0.194i)T + (-1.74 + 2.44i)T^{2}
5 1+(1.63−0.156i)T+(4.90−0.946i)T2 1 + (1.63 - 0.156i)T + (4.90 - 0.946i)T^{2}
11 1+(1.66−0.576i)T+(8.64−6.79i)T2 1 + (1.66 - 0.576i)T + (8.64 - 6.79i)T^{2}
13 1+(−0.504−1.71i)T+(−10.9+7.02i)T2 1 + (-0.504 - 1.71i)T + (-10.9 + 7.02i)T^{2}
17 1+(1.46+0.588i)T+(12.3+11.7i)T2 1 + (1.46 + 0.588i)T + (12.3 + 11.7i)T^{2}
19 1+(5.01−2.00i)T+(13.7−13.1i)T2 1 + (5.01 - 2.00i)T + (13.7 - 13.1i)T^{2}
29 1+(−0.374−2.60i)T+(−27.8+8.17i)T2 1 + (-0.374 - 2.60i)T + (-27.8 + 8.17i)T^{2}
31 1+(−2.61+0.124i)T+(30.8−2.94i)T2 1 + (-2.61 + 0.124i)T + (30.8 - 2.94i)T^{2}
37 1+(0.803+0.571i)T+(12.1+34.9i)T2 1 + (0.803 + 0.571i)T + (12.1 + 34.9i)T^{2}
41 1+(8.66+3.95i)T+(26.8+30.9i)T2 1 + (8.66 + 3.95i)T + (26.8 + 30.9i)T^{2}
43 1+(0.488−0.760i)T+(−17.8−39.1i)T2 1 + (0.488 - 0.760i)T + (-17.8 - 39.1i)T^{2}
47 1+(3.61−2.08i)T+(23.5−40.7i)T2 1 + (3.61 - 2.08i)T + (23.5 - 40.7i)T^{2}
53 1+(−0.118−0.124i)T+(−2.52+52.9i)T2 1 + (-0.118 - 0.124i)T + (-2.52 + 52.9i)T^{2}
59 1+(−7.99−1.93i)T+(52.4+27.0i)T2 1 + (-7.99 - 1.93i)T + (52.4 + 27.0i)T^{2}
61 1+(8.21+4.23i)T+(35.3+49.6i)T2 1 + (8.21 + 4.23i)T + (35.3 + 49.6i)T^{2}
67 1+(−1.93−10.0i)T+(−62.2+24.9i)T2 1 + (-1.93 - 10.0i)T + (-62.2 + 24.9i)T^{2}
71 1+(0.969+1.11i)T+(−10.1+70.2i)T2 1 + (0.969 + 1.11i)T + (-10.1 + 70.2i)T^{2}
73 1+(−6.13+7.80i)T+(−17.2−70.9i)T2 1 + (-6.13 + 7.80i)T + (-17.2 - 70.9i)T^{2}
79 1+(−11.2+11.8i)T+(−3.75−78.9i)T2 1 + (-11.2 + 11.8i)T + (-3.75 - 78.9i)T^{2}
83 1+(−5.57−12.2i)T+(−54.3+62.7i)T2 1 + (-5.57 - 12.2i)T + (-54.3 + 62.7i)T^{2}
89 1+(−0.563+11.8i)T+(−88.5−8.45i)T2 1 + (-0.563 + 11.8i)T + (-88.5 - 8.45i)T^{2}
97 1+(5.49−12.0i)T+(−63.5−73.3i)T2 1 + (5.49 - 12.0i)T + (-63.5 - 73.3i)T^{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

−12.41648009664203122071929379918, −11.89628365617482802058154433133, −10.97960341853672263135264305138, −10.11561712160910298453516764894, −8.555897290211869544490335457809, −7.64307113124653144480960568283, −6.54659136329791103850084160655, −4.57264644693304889007233773719, −3.71494101681311483207314884224, −1.97395979102939318204778252592, 2.03141245114383833722645573082, 4.33502077107473761622042738616, 5.29723124836754827049521990751, 6.67410017228748357292620100770, 7.902789577318074033455929123881, 8.177196881908712612886579992058, 10.23625865784917470878142188007, 10.96897214102866627283886387820, 11.83285522604387892835482121155, 13.23130316959584708858975130958

Graph of the ZZ-function along the critical line