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Properties

Label 2-161-161.104-c0-0-0
Degree $2$
Conductor $161$
Sign $0.947 + 0.320i$
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  + (0.698 − 0.449i)2-s + (−0.128 + 0.281i)4-s + (−0.654 − 0.755i)7-s + (0.154 + 1.07i)8-s + (−0.959 − 0.281i)9-s + (−0.239 − 0.153i)11-s + (−0.797 − 0.234i)14-s + (0.389 + 0.449i)16-s + (−0.797 + 0.234i)18-s − 0.236·22-s + (−0.959 + 0.281i)23-s + (0.841 − 0.540i)25-s + (0.297 − 0.0872i)28-s + (0.698 + 1.53i)29-s + (−0.570 − 0.167i)32-s + ⋯
L(s)  = 1  + (0.698 − 0.449i)2-s + (−0.128 + 0.281i)4-s + (−0.654 − 0.755i)7-s + (0.154 + 1.07i)8-s + (−0.959 − 0.281i)9-s + (−0.239 − 0.153i)11-s + (−0.797 − 0.234i)14-s + (0.389 + 0.449i)16-s + (−0.797 + 0.234i)18-s − 0.236·22-s + (−0.959 + 0.281i)23-s + (0.841 − 0.540i)25-s + (0.297 − 0.0872i)28-s + (0.698 + 1.53i)29-s + (−0.570 − 0.167i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.947 + 0.320i$
Analytic conductor: \(0.0803494\)
Root analytic conductor: \(0.283459\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :0),\ 0.947 + 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7892711258\)
\(L(\frac12)\) \(\approx\) \(0.7892711258\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (0.959 - 0.281i)T \)
good2 \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \)
3 \( 1 + (0.959 + 0.281i)T^{2} \)
5 \( 1 + (-0.841 + 0.540i)T^{2} \)
11 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (0.142 + 0.989i)T^{2} \)
17 \( 1 + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.959 - 0.281i)T^{2} \)
37 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
41 \( 1 + (-0.841 + 0.540i)T^{2} \)
43 \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.959 - 0.281i)T^{2} \)
67 \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
97 \( 1 + (-0.841 + 0.540i)T^{2} \)
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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

−13.00123841674341366897227728483, −12.23917667169982410648489082480, −11.25832114909799839935705014639, −10.30485347322473292214345155944, −8.942439123758847158387470289726, −7.954197474781591509738456373365, −6.56142290725055414226881583428, −5.23084737949748298265897691008, −3.87005284387443551242482745015, −2.84042916572077859047926445564, 2.82436428438106411911833168692, 4.49684778315265612994674574284, 5.73818220175323534355701108509, 6.35672497454848434603325579429, 7.907331391170097635124771794478, 9.162074868632621403998135519093, 10.05306667164998625694276148263, 11.32153121039703898262483719262, 12.45339961642386166593797889354, 13.26367825065869601184848425155

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