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Properties

Label 1-161-161.31-r1-0-0
Degree $1$
Conductor $161$
Sign $0.769 - 0.639i$
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  + (0.723 − 0.690i)2-s + (−0.981 − 0.189i)3-s + (0.0475 − 0.998i)4-s + (0.786 + 0.618i)5-s + (−0.841 + 0.540i)6-s + (−0.654 − 0.755i)8-s + (0.928 + 0.371i)9-s + (0.995 − 0.0950i)10-s + (0.723 + 0.690i)11-s + (−0.235 + 0.971i)12-s + (−0.415 + 0.909i)13-s + (−0.654 − 0.755i)15-s + (−0.995 − 0.0950i)16-s + (0.888 + 0.458i)17-s + (0.928 − 0.371i)18-s + (0.888 − 0.458i)19-s + ⋯
L(s)  = 1  + (0.723 − 0.690i)2-s + (−0.981 − 0.189i)3-s + (0.0475 − 0.998i)4-s + (0.786 + 0.618i)5-s + (−0.841 + 0.540i)6-s + (−0.654 − 0.755i)8-s + (0.928 + 0.371i)9-s + (0.995 − 0.0950i)10-s + (0.723 + 0.690i)11-s + (−0.235 + 0.971i)12-s + (−0.415 + 0.909i)13-s + (−0.654 − 0.755i)15-s + (−0.995 − 0.0950i)16-s + (0.888 + 0.458i)17-s + (0.928 − 0.371i)18-s + (0.888 − 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.769 - 0.639i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ 0.769 - 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.237479957 - 0.8082235813i\)
\(L(\frac12)\) \(\approx\) \(2.237479957 - 0.8082235813i\)
\(L(1)\) \(\approx\) \(1.380210485 - 0.4551290470i\)
\(L(1)\) \(\approx\) \(1.380210485 - 0.4551290470i\)

Euler product

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

−27.48930811145645918569735668380, −26.80495309557083489640668171372, −25.19260445561671339602956499064, −24.795672240788679163970869373226, −23.73876109258296657820026937115, −22.78886908544152199523885502465, −21.92590920382765154283468932479, −21.27224005661703429497735997848, −20.15994851632301476877493242848, −18.26756907607919257560740291981, −17.446856032882190067652264849757, −16.58831980858594194782862115264, −15.99876593657885453317990002868, −14.58014408509954250560628963154, −13.59630556542598116773892672401, −12.47563422944608684483925400743, −11.81412416411386634629516894934, −10.30963378929040372486909480755, −9.10360173004694732702628197469, −7.67187697704223516311790808893, −6.32273587465494222479030586675, −5.54718133744390462520209674199, −4.72719138554821615750825355863, −3.207138798103295021441610961058, −1.00514216245451359357876925031, 1.18967165610726553190644586370, 2.36195186227083810881086821052, 4.06265786573299835475878370064, 5.243424192533269248623754664942, 6.27794217333893064238224222259, 7.10652128466727214116525010444, 9.5873458192699397109650581931, 10.14524451503371189697445683999, 11.432204957235064841609036783982, 12.05369844193511687316666941735, 13.24217829545056594029413153230, 14.19904999551374400086778728319, 15.181284387436381349146950529015, 16.65130422602909043224129706430, 17.67342205105773769078456843327, 18.61638657895012091395685594779, 19.51451784622747864582910063461, 20.95019803497588381744881606038, 21.801684049044320852221801265502, 22.42696023887323061272000718932, 23.273035380471116019110951564620, 24.28126535139549840167400281080, 25.219420081957305014175663500062, 26.6636635056986441709972956223, 27.86560453293485264552975495166

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