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Properties

Label 1-161-161.88-r1-0-0
Degree $1$
Conductor $161$
Sign $-0.279 + 0.960i$
Analytic cond. $17.3018$
Root an. cond. $17.3018$
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  + (−0.327 − 0.945i)2-s + (−0.888 − 0.458i)3-s + (−0.786 + 0.618i)4-s + (0.995 + 0.0950i)5-s + (−0.142 + 0.989i)6-s + (0.841 + 0.540i)8-s + (0.580 + 0.814i)9-s + (−0.235 − 0.971i)10-s + (0.327 − 0.945i)11-s + (0.981 − 0.189i)12-s + (−0.959 − 0.281i)13-s + (−0.841 − 0.540i)15-s + (0.235 − 0.971i)16-s + (−0.928 + 0.371i)17-s + (0.580 − 0.814i)18-s + (−0.928 − 0.371i)19-s + ⋯
L(s)  = 1  + (−0.327 − 0.945i)2-s + (−0.888 − 0.458i)3-s + (−0.786 + 0.618i)4-s + (0.995 + 0.0950i)5-s + (−0.142 + 0.989i)6-s + (0.841 + 0.540i)8-s + (0.580 + 0.814i)9-s + (−0.235 − 0.971i)10-s + (0.327 − 0.945i)11-s + (0.981 − 0.189i)12-s + (−0.959 − 0.281i)13-s + (−0.841 − 0.540i)15-s + (0.235 − 0.971i)16-s + (−0.928 + 0.371i)17-s + (0.580 − 0.814i)18-s + (−0.928 − 0.371i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.279 + 0.960i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ -0.279 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06057209382 - 0.08070500995i\)
\(L(\frac12)\) \(\approx\) \(-0.06057209382 - 0.08070500995i\)
\(L(1)\) \(\approx\) \(0.4959003203 - 0.3167950098i\)
\(L(1)\) \(\approx\) \(0.4959003203 - 0.3167950098i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.327 - 0.945i)T \)
3 \( 1 + (-0.888 - 0.458i)T \)
5 \( 1 + (0.995 + 0.0950i)T \)
11 \( 1 + (0.327 - 0.945i)T \)
13 \( 1 + (-0.959 - 0.281i)T \)
17 \( 1 + (-0.928 + 0.371i)T \)
19 \( 1 + (-0.928 - 0.371i)T \)
29 \( 1 + (-0.142 + 0.989i)T \)
31 \( 1 + (0.0475 - 0.998i)T \)
37 \( 1 + (-0.580 - 0.814i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.723 - 0.690i)T \)
59 \( 1 + (0.235 + 0.971i)T \)
61 \( 1 + (0.888 - 0.458i)T \)
67 \( 1 + (-0.981 - 0.189i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (-0.786 + 0.618i)T \)
79 \( 1 + (-0.723 + 0.690i)T \)
83 \( 1 + (-0.415 + 0.909i)T \)
89 \( 1 + (-0.0475 - 0.998i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−28.18906490487486847036462462967, −27.16911977583248390945740658197, −26.34466263639402143232629449202, −25.25925811827888636288313546437, −24.48532884059664802480060529777, −23.40794676063304090535385798498, −22.43418999176970374055435446927, −21.798013995705466386660772025272, −20.51909545117111942495911706159, −19.04607867538685215659650341710, −17.75666922339909052645158713095, −17.38454611999800154360562596581, −16.5506726922845836858089648184, −15.364663461288722106608449694193, −14.52960437693605344797932268725, −13.26282794757185522492911740600, −12.11402798533637853499144250229, −10.48159819790316019663082104516, −9.78780019280057872277408231629, −8.87486463480045890168798004245, −7.06725423448679883289731857971, −6.357699408644839570640876442070, −5.15531649342248279979375878568, −4.388848937520733373025708452428, −1.795648960359250824903375418159, 0.046631738103965239133541149938, 1.51411673857556897435785326586, 2.65739244931991767028987842167, 4.50275387425563200450929913291, 5.6999784120419633955695130677, 6.901405536523564545471527424249, 8.425855463469226156943334380198, 9.631820163411731305429914281208, 10.68548903135886925575081683536, 11.39149572802560449387976000885, 12.7294422983849050865702819159, 13.2601458721838946075299648988, 14.490833209079218007196877304971, 16.4555946913951542866773115288, 17.29365331847927013408580755546, 17.88662482168625432209681904031, 18.95233443012836199875744225855, 19.7811381394114799814735856410, 21.25949470506945663259965778438, 21.93382458332618243750561140487, 22.535084785681453126203830368790, 23.938013698771008103202806975380, 24.854596526518778601277494489815, 26.10338089761717604847090901158, 27.12092515118274355222630612781

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