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Properties

Label 1-179-179.4-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: $\chi_{179} (4, \cdot )$
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

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

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