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Properties

Label 1-161-161.3-r1-0-0
Degree $1$
Conductor $161$
Sign $0.584 + 0.811i$
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  + (0.235 − 0.971i)2-s + (0.327 − 0.945i)3-s + (−0.888 − 0.458i)4-s + (−0.928 − 0.371i)5-s + (−0.841 − 0.540i)6-s + (−0.654 + 0.755i)8-s + (−0.786 − 0.618i)9-s + (−0.580 + 0.814i)10-s + (0.235 + 0.971i)11-s + (−0.723 + 0.690i)12-s + (−0.415 − 0.909i)13-s + (−0.654 + 0.755i)15-s + (0.580 + 0.814i)16-s + (−0.0475 + 0.998i)17-s + (−0.786 + 0.618i)18-s + (−0.0475 − 0.998i)19-s + ⋯
L(s)  = 1  + (0.235 − 0.971i)2-s + (0.327 − 0.945i)3-s + (−0.888 − 0.458i)4-s + (−0.928 − 0.371i)5-s + (−0.841 − 0.540i)6-s + (−0.654 + 0.755i)8-s + (−0.786 − 0.618i)9-s + (−0.580 + 0.814i)10-s + (0.235 + 0.971i)11-s + (−0.723 + 0.690i)12-s + (−0.415 − 0.909i)13-s + (−0.654 + 0.755i)15-s + (0.580 + 0.814i)16-s + (−0.0475 + 0.998i)17-s + (−0.786 + 0.618i)18-s + (−0.0475 − 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ 0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1468375956 - 0.07521505602i\)
\(L(\frac12)\) \(\approx\) \(-0.1468375956 - 0.07521505602i\)
\(L(1)\) \(\approx\) \(0.4951304244 - 0.6052041125i\)
\(L(1)\) \(\approx\) \(0.4951304244 - 0.6052041125i\)

Euler product

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

−27.80604911383236067590297918256, −26.97528735218797240820394441559, −26.69135103427757375268500750889, −25.54988534820178078664234338780, −24.52094634009587702506438625330, −23.49656397427989331196134272132, −22.55525178509645856798727604445, −21.822153696594410556133122990, −20.76183681214312311416923451561, −19.379959534321006843393938609423, −18.591953213490007902799829443330, −16.99616739657123054866308088467, −16.19931473929771983775996317327, −15.57972363071860953504468724582, −14.37375385342689847727038689474, −13.97397163255641988667282692299, −12.17517757179543950017961669356, −11.11617615805556686334021816287, −9.68204860559428331164861519741, −8.6541391481116710232826708677, −7.73601037272856737406118709628, −6.46995486291318023069444181263, −5.06214248989134601549896829770, −4.02521562544212881344040139286, −3.1131318982496271221136061216, 0.05658406167246527982777847307, 1.41718318189094537814179430917, 2.77761193489530014627306192341, 4.007096736245269460737675479020, 5.306871710902069712681193470605, 6.999953183716003477045356455331, 8.16790846131793681401292495967, 9.100301520033774731789186894346, 10.55340654437328499066130628697, 11.765908047962037537799857862920, 12.545523711077488444959755062150, 13.13821273438803729862314110503, 14.57630400787316302843326668250, 15.30282578063710079972598303318, 17.23100709285121294664493425102, 18.020284287429199189869879439194, 19.18693395292592867677671697965, 19.93167781845848370865127257954, 20.33180151988234162964888322285, 21.84968554934882483149179330524, 22.987360136597408608631801644393, 23.61526520115644444830435429938, 24.52699777426016865765590772470, 25.76401701796219956674929897731, 26.9521809925424152993957230609

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