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Properties

Label 1-179-179.45-r0-0-0
Degree $1$
Conductor $179$
Sign $0.567 - 0.823i$
Analytic cond. $0.831272$
Root an. cond. $0.831272$
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  + (0.997 − 0.0705i)2-s + (0.227 − 0.973i)3-s + (0.990 − 0.140i)4-s + (−0.949 + 0.312i)5-s + (0.158 − 0.987i)6-s + (0.880 + 0.474i)7-s + (0.977 − 0.210i)8-s + (−0.896 − 0.442i)9-s + (−0.925 + 0.378i)10-s + (0.489 − 0.871i)11-s + (0.0881 − 0.996i)12-s + (−0.192 − 0.981i)13-s + (0.911 + 0.411i)14-s + (0.0881 + 0.996i)15-s + (0.960 − 0.278i)16-s + (0.662 + 0.749i)17-s + ⋯
L(s)  = 1  + (0.997 − 0.0705i)2-s + (0.227 − 0.973i)3-s + (0.990 − 0.140i)4-s + (−0.949 + 0.312i)5-s + (0.158 − 0.987i)6-s + (0.880 + 0.474i)7-s + (0.977 − 0.210i)8-s + (−0.896 − 0.442i)9-s + (−0.925 + 0.378i)10-s + (0.489 − 0.871i)11-s + (0.0881 − 0.996i)12-s + (−0.192 − 0.981i)13-s + (0.911 + 0.411i)14-s + (0.0881 + 0.996i)15-s + (0.960 − 0.278i)16-s + (0.662 + 0.749i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.567 - 0.823i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.567 - 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.830043519 - 0.9609332684i\)
\(L(\frac12)\) \(\approx\) \(1.830043519 - 0.9609332684i\)
\(L(1)\) \(\approx\) \(1.726346225 - 0.5753822985i\)
\(L(1)\) \(\approx\) \(1.726346225 - 0.5753822985i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.997 - 0.0705i)T \)
3 \( 1 + (0.227 - 0.973i)T \)
5 \( 1 + (-0.949 + 0.312i)T \)
7 \( 1 + (0.880 + 0.474i)T \)
11 \( 1 + (0.489 - 0.871i)T \)
13 \( 1 + (-0.192 - 0.981i)T \)
17 \( 1 + (0.662 + 0.749i)T \)
19 \( 1 + (-0.783 + 0.621i)T \)
23 \( 1 + (-0.994 + 0.105i)T \)
29 \( 1 + (-0.458 - 0.888i)T \)
31 \( 1 + (-0.329 + 0.944i)T \)
37 \( 1 + (-0.458 + 0.888i)T \)
41 \( 1 + (-0.0529 + 0.998i)T \)
43 \( 1 + (0.804 + 0.593i)T \)
47 \( 1 + (-0.825 - 0.564i)T \)
53 \( 1 + (-0.635 - 0.772i)T \)
59 \( 1 + (0.295 + 0.955i)T \)
61 \( 1 + (0.362 + 0.932i)T \)
67 \( 1 + (0.977 + 0.210i)T \)
71 \( 1 + (-0.520 + 0.854i)T \)
73 \( 1 + (-0.688 - 0.725i)T \)
79 \( 1 + (0.713 - 0.700i)T \)
83 \( 1 + (-0.123 - 0.992i)T \)
89 \( 1 + (0.997 + 0.0705i)T \)
97 \( 1 + (0.0176 + 0.999i)T \)
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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

−27.68329198803427581529917054615, −26.472560460870266263895760002377, −25.60082828223971435750293733247, −24.32567485701152115603991209288, −23.597228680370055918413544448845, −22.69509792225126600821588827323, −21.758475068533500673670966758612, −20.671769963606134682412765310302, −20.27510871239247883011143444619, −19.23610885910351549538826046998, −17.26658501831289098345663372724, −16.44076867072616585700324594143, −15.55762406726894212315370993627, −14.59435324926357604550419024796, −14.08221546780783220077932233114, −12.44214545084633610075832123844, −11.53658474901851279803654477715, −10.789882822827049261479566397836, −9.32009465771403837662255771301, −7.961973290905357005900301293443, −7.01121821434058075282876092647, −5.21167676179269954311930070240, −4.36632401019370421897955172036, −3.787101012073314146683427418154, −2.07695967074139058195942780831, 1.49268454198210245371574442186, 2.8959605679928197070522645197, 3.92662557773645631142871490319, 5.52953032657227850539191572171, 6.45157006174539524187970563360, 7.86994102330415951331022767381, 8.2530130699915527069586060143, 10.58678296594949478114248439548, 11.65016182088405703270683004229, 12.188532639349433325957284378263, 13.2568406351424635520812149892, 14.60809882892664273122557009604, 14.790776074487380843901640876741, 16.20472434679702149091891138975, 17.49467190227723097143213231883, 18.80854300275284289573859214023, 19.46826534070766680141018960933, 20.41160250996924656259729443168, 21.52193081859667834428206379805, 22.611122477332472834585935784146, 23.488636277892606418841559971863, 24.21401793858631187156368529627, 24.88885048284133576757257719185, 25.91746124690241733763118531713, 27.34679162160689500311951943596

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