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Properties

Label 1-161-161.83-r0-0-0
Degree $1$
Conductor $161$
Sign $-0.767 - 0.641i$
Analytic cond. $0.747680$
Root an. cond. $0.747680$
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  + (0.841 − 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (−0.415 − 0.909i)6-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (−0.841 − 0.540i)11-s + (−0.841 − 0.540i)12-s + (0.654 − 0.755i)13-s + (0.142 + 0.989i)15-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)17-s + (−0.959 + 0.281i)18-s + (0.415 − 0.909i)19-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (−0.415 − 0.909i)6-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (−0.841 − 0.540i)11-s + (−0.841 − 0.540i)12-s + (0.654 − 0.755i)13-s + (0.142 + 0.989i)15-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)17-s + (−0.959 + 0.281i)18-s + (0.415 − 0.909i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.767 - 0.641i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ -0.767 - 0.641i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4962245912 - 1.366653710i\)
\(L(\frac12)\) \(\approx\) \(0.4962245912 - 1.366653710i\)
\(L(1)\) \(\approx\) \(1.002244289 - 0.9572596213i\)
\(L(1)\) \(\approx\) \(1.002244289 - 0.9572596213i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.841 - 0.540i)T \)
3 \( 1 + (0.142 - 0.989i)T \)
5 \( 1 + (-0.959 + 0.281i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (0.959 + 0.281i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 - T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.050575837720601917679106817945, −26.95496266823051644524376435920, −26.24885184756968659174850961241, −25.28478553462788332805433238831, −24.13412493696852990280565326731, −22.980791416504095288042956576104, −22.76135018665593533602268280360, −21.07578548786825363067737474112, −20.864269378724043365273050892856, −19.68829879706951304531100228726, −18.16779390309434100631837353025, −16.565272040088844180767942587548, −16.15562061935817250215807775664, −15.26741554604529822560903660736, −14.39408598409018119910582067206, −13.23435121195528185779688114586, −11.91366837542900223644976750658, −11.20179011154717772123674226985, −9.674829039126349076618435516952, −8.313081375978896403782076874380, −7.50909883108036793115038056401, −5.87251476830706071421275379185, −4.6983123952247174116134964408, −3.96650477176224259057506299910, −2.75789175028278286738034309466, 0.96949418547801081312033531629, 2.72893533083647558533334096405, 3.54199149067724043580028063225, 5.21656091314060723781356100778, 6.376651565733203032838017799748, 7.53693921858575118129968633750, 8.59261965954342972695601896913, 10.559614436107936380409249588314, 11.29523440605644516225242590316, 12.40916183069792469194244120598, 13.114713210003131279804768312509, 14.1606938611292412067161115485, 15.20553502348221873185484741356, 16.12459598122165562327613716372, 17.94627384239794541471961658661, 18.79837870176097017310293818613, 19.633567082940678927906282433, 20.35794935459825602001898538085, 21.61566675608287448832810600729, 22.79589437482988420920789456086, 23.5845691382431166235439914865, 24.03028015512507121986116147475, 25.24141212552251016552682412450, 26.334703037976152254701801615020, 27.73291628681128152298861039442

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