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Properties

Label 1-177-177.2-r0-0-0
Degree $1$
Conductor $177$
Sign $0.603 - 0.797i$
Analytic cond. $0.821984$
Root an. cond. $0.821984$
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  + (−0.994 − 0.108i)2-s + (0.976 + 0.214i)4-s + (−0.796 − 0.605i)5-s + (−0.370 + 0.928i)7-s + (−0.947 − 0.319i)8-s + (0.725 + 0.687i)10-s + (0.907 − 0.419i)11-s + (0.161 − 0.986i)13-s + (0.468 − 0.883i)14-s + (0.907 + 0.419i)16-s + (0.370 + 0.928i)17-s + (−0.561 − 0.827i)19-s + (−0.647 − 0.762i)20-s + (−0.947 + 0.319i)22-s + (0.0541 − 0.998i)23-s + ⋯
L(s)  = 1  + (−0.994 − 0.108i)2-s + (0.976 + 0.214i)4-s + (−0.796 − 0.605i)5-s + (−0.370 + 0.928i)7-s + (−0.947 − 0.319i)8-s + (0.725 + 0.687i)10-s + (0.907 − 0.419i)11-s + (0.161 − 0.986i)13-s + (0.468 − 0.883i)14-s + (0.907 + 0.419i)16-s + (0.370 + 0.928i)17-s + (−0.561 − 0.827i)19-s + (−0.647 − 0.762i)20-s + (−0.947 + 0.319i)22-s + (0.0541 − 0.998i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.603 - 0.797i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ 0.603 - 0.797i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5456801104 - 0.2712825898i\)
\(L(\frac12)\) \(\approx\) \(0.5456801104 - 0.2712825898i\)
\(L(1)\) \(\approx\) \(0.6213012971 - 0.1213135860i\)
\(L(1)\) \(\approx\) \(0.6213012971 - 0.1213135860i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.994 - 0.108i)T \)
5 \( 1 + (-0.796 - 0.605i)T \)
7 \( 1 + (-0.370 + 0.928i)T \)
11 \( 1 + (0.907 - 0.419i)T \)
13 \( 1 + (0.161 - 0.986i)T \)
17 \( 1 + (0.370 + 0.928i)T \)
19 \( 1 + (-0.561 - 0.827i)T \)
23 \( 1 + (0.0541 - 0.998i)T \)
29 \( 1 + (0.994 - 0.108i)T \)
31 \( 1 + (0.561 - 0.827i)T \)
37 \( 1 + (0.947 - 0.319i)T \)
41 \( 1 + (-0.0541 - 0.998i)T \)
43 \( 1 + (-0.907 - 0.419i)T \)
47 \( 1 + (0.796 - 0.605i)T \)
53 \( 1 + (0.725 - 0.687i)T \)
61 \( 1 + (0.994 + 0.108i)T \)
67 \( 1 + (0.947 + 0.319i)T \)
71 \( 1 + (-0.796 + 0.605i)T \)
73 \( 1 + (-0.468 + 0.883i)T \)
79 \( 1 + (0.647 + 0.762i)T \)
83 \( 1 + (-0.856 + 0.515i)T \)
89 \( 1 + (-0.994 + 0.108i)T \)
97 \( 1 + (-0.468 - 0.883i)T \)
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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

−27.1590183400591730415011118525, −26.89311650843959688737646046834, −25.75586959613452250635996527305, −24.991212871969864696437178233970, −23.53039170747467765403258293370, −23.13614047282391391995724911341, −21.63601893296365128296723064330, −20.37060451546608313176663006037, −19.579257582948633604769103688693, −18.92651719458037204602500087759, −17.816812727138361650087424462925, −16.71063019058988314079398304896, −16.066815258803799432247799015490, −14.84520043385232615580548187479, −13.95807699653111193529574231662, −12.11353277815099975127078708342, −11.40220829939125523655403477652, −10.28631296059621620295849895553, −9.40115578883199222961705866446, −8.058866879618675519122636080142, −7.06581996089188004682083910453, −6.43417311987808309452805913279, −4.28286427635382101848905941687, −3.07388746512661983888255651664, −1.293947208334776061666588082247, 0.791759571027540452366273583960, 2.55623804782521430460308786376, 3.88021507691024724208910598976, 5.68945032060159101984572839902, 6.80415473301305841163493290157, 8.373575419542456641255432371864, 8.62916339926986833298394835513, 9.963457957321012252247518844043, 11.19642310610768826578531104272, 12.125879208913156686143611496706, 12.88677828809251032625324869866, 14.920420760284716738683912623037, 15.58379198770776792011810380108, 16.58383077919707159425334049984, 17.4285614286724734589794289664, 18.70065098029196472609164055858, 19.41794690884597417680958085594, 20.145256371931281985804858802571, 21.293603517625893370715385808179, 22.305979523946692960848975433153, 23.67796644555192686385959293393, 24.698125503774409180516039808776, 25.284267626030472900135623506255, 26.44707544883666404419841261053, 27.48349375703368052465293276485

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