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Properties

Label 1-179-179.110-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.245 - 0.969i$
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  + (0.662 − 0.749i)2-s + (−0.925 + 0.378i)3-s + (−0.123 − 0.992i)4-s + (−0.783 + 0.621i)5-s + (−0.329 + 0.944i)6-s + (0.938 − 0.345i)7-s + (−0.825 − 0.564i)8-s + (0.713 − 0.700i)9-s + (−0.0529 + 0.998i)10-s + (0.990 − 0.140i)11-s + (0.489 + 0.871i)12-s + (−0.688 − 0.725i)13-s + (0.362 − 0.932i)14-s + (0.489 − 0.871i)15-s + (−0.969 + 0.244i)16-s + (−0.737 − 0.675i)17-s + ⋯
L(s)  = 1  + (0.662 − 0.749i)2-s + (−0.925 + 0.378i)3-s + (−0.123 − 0.992i)4-s + (−0.783 + 0.621i)5-s + (−0.329 + 0.944i)6-s + (0.938 − 0.345i)7-s + (−0.825 − 0.564i)8-s + (0.713 − 0.700i)9-s + (−0.0529 + 0.998i)10-s + (0.990 − 0.140i)11-s + (0.489 + 0.871i)12-s + (−0.688 − 0.725i)13-s + (0.362 − 0.932i)14-s + (0.489 − 0.871i)15-s + (−0.969 + 0.244i)16-s + (−0.737 − 0.675i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.245 - 0.969i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.245 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6309903898 - 0.8109864892i\)
\(L(\frac12)\) \(\approx\) \(0.6309903898 - 0.8109864892i\)
\(L(1)\) \(\approx\) \(0.8955752828 - 0.4896970994i\)
\(L(1)\) \(\approx\) \(0.8955752828 - 0.4896970994i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.662 - 0.749i)T \)
3 \( 1 + (-0.925 + 0.378i)T \)
5 \( 1 + (-0.783 + 0.621i)T \)
7 \( 1 + (0.938 - 0.345i)T \)
11 \( 1 + (0.990 - 0.140i)T \)
13 \( 1 + (-0.688 - 0.725i)T \)
17 \( 1 + (-0.737 - 0.675i)T \)
19 \( 1 + (-0.192 - 0.981i)T \)
23 \( 1 + (0.295 - 0.955i)T \)
29 \( 1 + (0.844 + 0.535i)T \)
31 \( 1 + (-0.635 - 0.772i)T \)
37 \( 1 + (0.844 - 0.535i)T \)
41 \( 1 + (0.804 + 0.593i)T \)
43 \( 1 + (0.227 + 0.973i)T \)
47 \( 1 + (0.607 + 0.794i)T \)
53 \( 1 + (-0.394 - 0.918i)T \)
59 \( 1 + (-0.896 + 0.442i)T \)
61 \( 1 + (-0.261 + 0.965i)T \)
67 \( 1 + (-0.825 + 0.564i)T \)
71 \( 1 + (0.960 + 0.278i)T \)
73 \( 1 + (-0.949 - 0.312i)T \)
79 \( 1 + (-0.994 - 0.105i)T \)
83 \( 1 + (0.0881 - 0.996i)T \)
89 \( 1 + (0.662 + 0.749i)T \)
97 \( 1 + (0.977 - 0.210i)T \)
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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

−27.375171549434855074050387739698, −27.00461168251676292742672744609, −25.172346356392581261233177284, −24.52705481821980597380252541828, −23.806914361805714155889569126675, −23.146004930161659203268625400, −21.97966755214703795807654530055, −21.34452279211734909607975524849, −19.92313395332947558842804282460, −18.71452828905531219080858302659, −17.33853484296175271538178991039, −17.02373440652596451305774601498, −15.87060074148418673810208864142, −14.93494855930422469378712460841, −13.84306859240409109833234355068, −12.43118991198531356631010394673, −12.010741921137130839681182220711, −11.11875417789168564875950461771, −9.06074519094166107289369923595, −7.96117748956419650688544703280, −7.040968462435131276179971778360, −5.86762171311757518547305615400, −4.752097512005739264626474403670, −4.0613754817529695525648003076, −1.73078027558303119854297115126, 0.80568100793783570004883890913, 2.74371366080334524747966292333, 4.24444237931941789242432684583, 4.75458882078787051912588107446, 6.2500441431937318255697075779, 7.30428410063704730322905671139, 9.14226238554362285705747041517, 10.50123342413201325971931633765, 11.18972143493289793519210702089, 11.77785831699667524954854781003, 12.88508841752960329496086456466, 14.44019215970506175667291794374, 14.97944056067744810385802428553, 16.11736822698591616192813694215, 17.53892688229400802678531257323, 18.277580515234965231466570409421, 19.59918935619530050261993555043, 20.32759152121601864444534178807, 21.57531876141176706794880208782, 22.32672691358908257535292156199, 22.93828091543691815793443366940, 23.95309932651021313049996678670, 24.6143658135242272591762580736, 26.73341413372045841875331842624, 27.31533365413415901735995476802

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