Properties

Label 2-161-161.10-c1-0-7
Degree $2$
Conductor $161$
Sign $0.866 + 0.499i$
Analytic cond. $1.28559$
Root an. cond. $1.13383$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.866 + 0.499i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :1/2),\ 0.866 + 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37085 - 0.366550i\)
\(L(\frac12)\) \(\approx\) \(1.37085 - 0.366550i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.63 - 0.196i)T \)
23 \( 1 + (4.30 + 2.11i)T \)
good2 \( 1 + (-0.318 + 0.919i)T + (-1.57 - 1.23i)T^{2} \)
3 \( 1 + (-0.100 - 0.194i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (1.63 - 0.156i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (1.66 - 0.576i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.504 - 1.71i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (1.46 + 0.588i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (5.01 - 2.00i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.374 - 2.60i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-2.61 + 0.124i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (0.803 + 0.571i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (8.66 + 3.95i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (0.488 - 0.760i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (3.61 - 2.08i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.118 - 0.124i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-7.99 - 1.93i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (8.21 + 4.23i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-1.93 - 10.0i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (0.969 + 1.11i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-6.13 + 7.80i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-11.2 + 11.8i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-5.57 - 12.2i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.563 + 11.8i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (5.49 - 12.0i)T + (-63.5 - 73.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line