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Properties

Label 2-161-161.41-c0-0-0
Degree $2$
Conductor $161$
Sign $0.918 + 0.394i$
Analytic cond. $0.0803494$
Root an. cond. $0.283459$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 − 0.474i)2-s + (1.54 + 0.989i)4-s + (0.415 + 0.909i)7-s + (−0.915 − 1.05i)8-s + (−0.142 − 0.989i)9-s + (1.25 − 0.368i)11-s + (−0.239 − 1.66i)14-s + (0.216 + 0.474i)16-s + (−0.239 + 1.66i)18-s − 2.20·22-s + (−0.142 + 0.989i)23-s + (−0.959 − 0.281i)25-s + (−0.260 + 1.81i)28-s + (−1.61 + 1.03i)29-s + (0.0741 + 0.515i)32-s + ⋯
L(s)  = 1  + (−1.61 − 0.474i)2-s + (1.54 + 0.989i)4-s + (0.415 + 0.909i)7-s + (−0.915 − 1.05i)8-s + (−0.142 − 0.989i)9-s + (1.25 − 0.368i)11-s + (−0.239 − 1.66i)14-s + (0.216 + 0.474i)16-s + (−0.239 + 1.66i)18-s − 2.20·22-s + (−0.142 + 0.989i)23-s + (−0.959 − 0.281i)25-s + (−0.260 + 1.81i)28-s + (−1.61 + 1.03i)29-s + (0.0741 + 0.515i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.918 + 0.394i$
Analytic conductor: \(0.0803494\)
Root analytic conductor: \(0.283459\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :0),\ 0.918 + 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3523491287\)
\(L(\frac12)\) \(\approx\) \(0.3523491287\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
good2 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
3 \( 1 + (0.142 + 0.989i)T^{2} \)
5 \( 1 + (0.959 + 0.281i)T^{2} \)
11 \( 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.654 + 0.755i)T^{2} \)
17 \( 1 + (-0.415 + 0.909i)T^{2} \)
19 \( 1 + (-0.415 - 0.909i)T^{2} \)
29 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
31 \( 1 + (0.142 - 0.989i)T^{2} \)
37 \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \)
41 \( 1 + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.797 + 1.74i)T + (-0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.654 + 0.755i)T^{2} \)
61 \( 1 + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (0.142 + 0.989i)T^{2} \)
97 \( 1 + (0.959 + 0.281i)T^{2} \)
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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

−12.50303801618415720663132915671, −11.65638061600290249722424621265, −11.15569023652201861751069073769, −9.613778795921184639355764832904, −9.175134633637133832528826241192, −8.298777730368382673725443069970, −7.06229313422605607261110702157, −5.79723419465515055535456798012, −3.52915002027807610006532954174, −1.74005362232392012658644655755, 1.74159903113150749498612699029, 4.31944668592089365522919906680, 6.14758628435401785003598347403, 7.32237786583621198437944673817, 7.951079035345400667434988397137, 9.080883522491623563615332307114, 10.00954352827088189901015286801, 10.87036690463096887745986543759, 11.69298138552481313683844488742, 13.34288141222982661684503449027

Graph of the $Z$-function along the critical line