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Properties

Label 1-161-161.136-r0-0-0
Degree $1$
Conductor $161$
Sign $-0.0815 - 0.996i$
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  + (−0.995 − 0.0950i)2-s + (−0.723 − 0.690i)3-s + (0.981 + 0.189i)4-s + (−0.888 + 0.458i)5-s + (0.654 + 0.755i)6-s + (−0.959 − 0.281i)8-s + (0.0475 + 0.998i)9-s + (0.928 − 0.371i)10-s + (0.995 − 0.0950i)11-s + (−0.580 − 0.814i)12-s + (0.142 − 0.989i)13-s + (0.959 + 0.281i)15-s + (0.928 + 0.371i)16-s + (−0.327 + 0.945i)17-s + (0.0475 − 0.998i)18-s + (−0.327 − 0.945i)19-s + ⋯
L(s)  = 1  + (−0.995 − 0.0950i)2-s + (−0.723 − 0.690i)3-s + (0.981 + 0.189i)4-s + (−0.888 + 0.458i)5-s + (0.654 + 0.755i)6-s + (−0.959 − 0.281i)8-s + (0.0475 + 0.998i)9-s + (0.928 − 0.371i)10-s + (0.995 − 0.0950i)11-s + (−0.580 − 0.814i)12-s + (0.142 − 0.989i)13-s + (0.959 + 0.281i)15-s + (0.928 + 0.371i)16-s + (−0.327 + 0.945i)17-s + (0.0475 − 0.998i)18-s + (−0.327 − 0.945i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0815 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0815 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.0815 - 0.996i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ -0.0815 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2745789834 - 0.2979490508i\)
\(L(\frac12)\) \(\approx\) \(0.2745789834 - 0.2979490508i\)
\(L(1)\) \(\approx\) \(0.4576732473 - 0.1538113325i\)
\(L(1)\) \(\approx\) \(0.4576732473 - 0.1538113325i\)

Euler product

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

−27.74036434296809182382614594429, −27.31027323535749743713481143988, −26.510928831918552504449354551101, −25.25010243918114754501347311086, −24.16083151894981002190208681085, −23.34683356533391496730468713135, −22.191450919662980655268933339751, −20.94257562017645089675524405120, −20.18045110167918685414797864570, −19.12433733595367986098330887133, −18.10867623238777158000600578319, −16.81103308944456214248133128628, −16.46444240305240108670531414099, −15.482067934236538200663945266362, −14.452505869202351627903703824907, −12.24671865631844776178419124038, −11.66626483158079225778910493501, −10.751827993359150168927560318750, −9.42037058013615129916636313000, −8.77332321543281809003490627474, −7.272156451621687440186051737088, −6.273168037022245064345361881, −4.755588230732013759871360176714, −3.537385353684328299179232018078, −1.3059212892926127713960163271, 0.574130428886037437041555104960, 2.22421344364273220874776297593, 3.84916409392407975166180265114, 5.876444277551218470162165659662, 6.881986890401253904767311830, 7.747360178237771600154151434050, 8.790461475861589150933148605901, 10.43269245444688163012399868191, 11.19890151953836363599045067867, 11.99489256147380427706327247280, 13.06152906309910498938665463211, 14.866470801228553247968362679346, 15.75822022662905136070583386375, 16.97797981828760725060457579859, 17.61568791292068856755235982967, 18.71397958824536079592023446330, 19.42438349781860246362202282061, 20.1663850444177844576333128171, 21.83618638678019752512889966901, 22.704294299046509847507936389795, 23.87505569596736260040910942258, 24.59547104004471871981254053474, 25.68331170343755101187270388644, 26.73585902688445034197569899413, 27.864120602426276807093420431522

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