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Properties

Label 1-161-161.17-r0-0-0
Degree $1$
Conductor $161$
Sign $0.676 + 0.736i$
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  + (0.981 − 0.189i)2-s + (−0.0475 + 0.998i)3-s + (0.928 − 0.371i)4-s + (0.580 + 0.814i)5-s + (0.142 + 0.989i)6-s + (0.841 − 0.540i)8-s + (−0.995 − 0.0950i)9-s + (0.723 + 0.690i)10-s + (−0.981 − 0.189i)11-s + (0.327 + 0.945i)12-s + (0.959 − 0.281i)13-s + (−0.841 + 0.540i)15-s + (0.723 − 0.690i)16-s + (−0.786 + 0.618i)17-s + (−0.995 + 0.0950i)18-s + (−0.786 − 0.618i)19-s + ⋯
L(s)  = 1  + (0.981 − 0.189i)2-s + (−0.0475 + 0.998i)3-s + (0.928 − 0.371i)4-s + (0.580 + 0.814i)5-s + (0.142 + 0.989i)6-s + (0.841 − 0.540i)8-s + (−0.995 − 0.0950i)9-s + (0.723 + 0.690i)10-s + (−0.981 − 0.189i)11-s + (0.327 + 0.945i)12-s + (0.959 − 0.281i)13-s + (−0.841 + 0.540i)15-s + (0.723 − 0.690i)16-s + (−0.786 + 0.618i)17-s + (−0.995 + 0.0950i)18-s + (−0.786 − 0.618i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.676 + 0.736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.861960447 + 0.8175430628i\)
\(L(\frac12)\) \(\approx\) \(1.861960447 + 0.8175430628i\)
\(L(1)\) \(\approx\) \(1.755931241 + 0.4728348364i\)
\(L(1)\) \(\approx\) \(1.755931241 + 0.4728348364i\)

Euler product

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

−28.13516225317251443734165316738, −26.16078980618793318319559741560, −25.37104281173221865262511297449, −24.652070170549280246726106374792, −23.700714430329598779771096964509, −23.16758236422882583902249627581, −21.862263943003725023887063986348, −20.75037287968769932241862779708, −20.2028781640334526087859423212, −18.75460216413068601442197622863, −17.67953127392684003871426288432, −16.654885881668806549023073412726, −15.69279820015360398721432977843, −14.27066013843948242802076515808, −13.331088959323825846898920697387, −12.88677549579839833129279846856, −11.80780832749592107092283131398, −10.64388137832799455504816074091, −8.78561370718685155065843106047, −7.77486330782556256966282448956, −6.48548569383280075231902085690, −5.6533881934571966140053653526, −4.4893774402032151229406118714, −2.70660355735587875660586516672, −1.58594145598000745239885322231, 2.32180887835320183682175452161, 3.31906369445368436649244082124, 4.52964096571953050275420210829, 5.736101833112127395186242615462, 6.54779928993005747248184670556, 8.318308582802923123773219491270, 9.955467444974958761970057921864, 10.728217102352678192618023067780, 11.38656654969411643253112514460, 13.12393186485111799594320527494, 13.803536920145571674485015955316, 15.145501209903655158443211825634, 15.445181671053749787820908986220, 16.75292106538820087420298763114, 18.00028615823675909044990461239, 19.32574487063323912024857157333, 20.51272616101438563260264597246, 21.35749739674923465875948352440, 21.88807776246999642090463569394, 22.93073972646362723190417199790, 23.61909636795229192517570769686, 25.14017681813943226753831594164, 25.96193612528273644231504951832, 26.73016703902022110265901563377, 28.322881436986953077843949634412

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