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Properties

Label 1-161-161.131-r1-0-0
Degree $1$
Conductor $161$
Sign $0.651 + 0.758i$
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  + (−0.786 + 0.618i)2-s + (−0.580 − 0.814i)3-s + (0.235 − 0.971i)4-s + (−0.981 − 0.189i)5-s + (0.959 + 0.281i)6-s + (0.415 + 0.909i)8-s + (−0.327 + 0.945i)9-s + (0.888 − 0.458i)10-s + (−0.786 − 0.618i)11-s + (−0.928 + 0.371i)12-s + (−0.841 − 0.540i)13-s + (0.415 + 0.909i)15-s + (−0.888 − 0.458i)16-s + (−0.723 + 0.690i)17-s + (−0.327 − 0.945i)18-s + (−0.723 − 0.690i)19-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (−0.580 − 0.814i)3-s + (0.235 − 0.971i)4-s + (−0.981 − 0.189i)5-s + (0.959 + 0.281i)6-s + (0.415 + 0.909i)8-s + (−0.327 + 0.945i)9-s + (0.888 − 0.458i)10-s + (−0.786 − 0.618i)11-s + (−0.928 + 0.371i)12-s + (−0.841 − 0.540i)13-s + (0.415 + 0.909i)15-s + (−0.888 − 0.458i)16-s + (−0.723 + 0.690i)17-s + (−0.327 − 0.945i)18-s + (−0.723 − 0.690i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.651 + 0.758i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ 0.651 + 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2717495151 + 0.1248061849i\)
\(L(\frac12)\) \(\approx\) \(0.2717495151 + 0.1248061849i\)
\(L(1)\) \(\approx\) \(0.4076285474 + 0.02770124320i\)
\(L(1)\) \(\approx\) \(0.4076285474 + 0.02770124320i\)

Euler product

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

−27.487444546145092505172705074858, −26.63185413176351311847043028436, −26.11117107682198238505224121971, −24.5443767397338164612208223141, −23.20197774401295114338117758518, −22.47729893553333484342285633789, −21.37514839085466963368694960981, −20.52565004594324316561771086078, −19.6055717692439930265154814572, −18.531655610687803817497706153106, −17.55816679681558136419209471035, −16.50537649183813580900127818119, −15.74936145006655419259775463245, −14.74177623374606879078456239853, −12.76333514417213208761295770588, −11.86623105913955773072405631240, −11.04319997466927233450213884243, −10.1248692683598057641888945005, −9.11719697587154125605336047558, −7.832482543936642889706147743270, −6.7665971186977593943317654613, −4.80178642765393689613092368091, −3.87508487565720429479647333035, −2.46020360015008838105034434358, −0.26255551132172365785771022098, 0.66541019263451213024763188324, 2.43137331531429285806289925993, 4.70542761166629336850785684379, 5.852686917678897748677236722229, 7.05591743924247809699910659436, 7.89119532208004265280121959158, 8.71726142221734552416651851555, 10.498410738272411229716552200837, 11.21946591283202967880722783791, 12.438898409623019298371284946976, 13.55092580427121282191687545841, 15.064865127831916272465110082560, 15.83466615644842968300361325412, 16.975132754233150072925555840380, 17.63753159629541904000793805086, 18.93607571345779474329084169897, 19.29878798814796147918990809470, 20.40145812756595559852373059810, 22.18900581629674718773468020211, 23.22455290406509559885014476357, 24.11299410589066666038922917863, 24.41545630593637752404149440560, 25.79272328592622425129198371053, 26.7591922474472735260177051967, 27.705822968138455036228731305888

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