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Properties

Label 1-161-161.153-r0-0-0
Degree $1$
Conductor $161$
Sign $0.903 - 0.427i$
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  + (−0.959 − 0.281i)2-s + (0.654 − 0.755i)3-s + (0.841 + 0.540i)4-s + (−0.142 + 0.989i)5-s + (−0.841 + 0.540i)6-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (0.959 − 0.281i)11-s + (0.959 − 0.281i)12-s + (−0.415 + 0.909i)13-s + (0.654 + 0.755i)15-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + (−0.142 + 0.989i)18-s + (0.841 + 0.540i)19-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.654 − 0.755i)3-s + (0.841 + 0.540i)4-s + (−0.142 + 0.989i)5-s + (−0.841 + 0.540i)6-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (0.959 − 0.281i)11-s + (0.959 − 0.281i)12-s + (−0.415 + 0.909i)13-s + (0.654 + 0.755i)15-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + (−0.142 + 0.989i)18-s + (0.841 + 0.540i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.903 - 0.427i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.903 - 0.427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9139575010 - 0.2053887413i\)
\(L(\frac12)\) \(\approx\) \(0.9139575010 - 0.2053887413i\)
\(L(1)\) \(\approx\) \(0.8790261880 - 0.1626017168i\)
\(L(1)\) \(\approx\) \(0.8790261880 - 0.1626017168i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.959 - 0.281i)T \)
3 \( 1 + (0.654 - 0.755i)T \)
5 \( 1 + (-0.142 + 0.989i)T \)
11 \( 1 + (0.959 - 0.281i)T \)
13 \( 1 + (-0.415 + 0.909i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (0.841 + 0.540i)T \)
29 \( 1 + (0.841 - 0.540i)T \)
31 \( 1 + (0.654 + 0.755i)T \)
37 \( 1 + (0.142 + 0.989i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (-0.415 + 0.909i)T \)
83 \( 1 + (-0.142 - 0.989i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
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

−27.81737827625498202780870161872, −26.99868706286174302095138202584, −25.997703653122761180069433316, −24.962150415016568152514194527329, −24.58630210615343872458827510526, −23.1218354672388661565373657670, −21.70923279876742340600136970786, −20.656647383340722206666068456280, −19.89890099660166785046449769624, −19.361019208665290099587241076867, −17.740196045215337830730547317817, −16.85776832411316533565861543772, −16.03900309700833000515784796323, −15.13012493730234234916934131163, −14.177277627726501153646458381269, −12.59623409436911826625024178651, −11.37630089226603445953553796457, −10.012073324711623724455290565, −9.37161704404627150707945480348, −8.3646296360550979247398706972, −7.53658809610794221954627296468, −5.77496478052996628799485483192, −4.57684591526623547127410431345, −2.99615511894681861555243242394, −1.30830130560265103165779606605, 1.36487910229301593162676692920, 2.68919260565593471939479484031, 3.64519094089369833556615734990, 6.33071754414588054889418035202, 7.087851796172723890834306379882, 8.01641176133848398230602438875, 9.21824449943150066136513359520, 10.11659065206970326607417587330, 11.673108106688062592041329597145, 12.06466117317955936506441736454, 13.86963296347696955926339809774, 14.55543708039509706154170991221, 15.85990886960689698932730662652, 17.12900120444409394761742678254, 18.12159290937300380135204569808, 19.01995803997444033092185927320, 19.41145169647191383950948941476, 20.58528657794784520767783429548, 21.64085676442345035921228412668, 22.865026190910797050437478142697, 24.205774298707511920476230324450, 25.06748017629422673342643949410, 25.87198496990437094602170146203, 26.77768831267526073110040127677, 27.38056010234673895148872014917

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