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Properties

Label 1-179-179.6-r1-0-0
Degree $1$
Conductor $179$
Sign $0.990 - 0.138i$
Analytic cond. $19.2362$
Root an. cond. $19.2362$
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  + (−0.760 − 0.648i)2-s + (0.662 + 0.749i)3-s + (0.158 + 0.987i)4-s + (−0.999 − 0.0352i)5-s + (−0.0176 − 0.999i)6-s + (−0.227 − 0.973i)7-s + (0.520 − 0.854i)8-s + (−0.123 + 0.992i)9-s + (0.737 + 0.675i)10-s + (0.394 + 0.918i)11-s + (−0.635 + 0.772i)12-s + (0.362 − 0.932i)13-s + (−0.458 + 0.888i)14-s + (−0.635 − 0.772i)15-s + (−0.949 + 0.312i)16-s + (−0.579 − 0.815i)17-s + ⋯
L(s)  = 1  + (−0.760 − 0.648i)2-s + (0.662 + 0.749i)3-s + (0.158 + 0.987i)4-s + (−0.999 − 0.0352i)5-s + (−0.0176 − 0.999i)6-s + (−0.227 − 0.973i)7-s + (0.520 − 0.854i)8-s + (−0.123 + 0.992i)9-s + (0.737 + 0.675i)10-s + (0.394 + 0.918i)11-s + (−0.635 + 0.772i)12-s + (0.362 − 0.932i)13-s + (−0.458 + 0.888i)14-s + (−0.635 − 0.772i)15-s + (−0.949 + 0.312i)16-s + (−0.579 − 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.990 - 0.138i$
Analytic conductor: \(19.2362\)
Root analytic conductor: \(19.2362\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (1:\ ),\ 0.990 - 0.138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.241907935 - 0.08640631509i\)
\(L(\frac12)\) \(\approx\) \(1.241907935 - 0.08640631509i\)
\(L(1)\) \(\approx\) \(0.8395060184 - 0.05941447706i\)
\(L(1)\) \(\approx\) \(0.8395060184 - 0.05941447706i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.760 - 0.648i)T \)
3 \( 1 + (0.662 + 0.749i)T \)
5 \( 1 + (-0.999 - 0.0352i)T \)
7 \( 1 + (-0.227 - 0.973i)T \)
11 \( 1 + (0.394 + 0.918i)T \)
13 \( 1 + (0.362 - 0.932i)T \)
17 \( 1 + (-0.579 - 0.815i)T \)
19 \( 1 + (0.911 + 0.411i)T \)
23 \( 1 + (-0.489 - 0.871i)T \)
29 \( 1 + (-0.0529 + 0.998i)T \)
31 \( 1 + (0.977 - 0.210i)T \)
37 \( 1 + (0.0529 + 0.998i)T \)
41 \( 1 + (0.863 - 0.505i)T \)
43 \( 1 + (0.997 - 0.0705i)T \)
47 \( 1 + (0.960 + 0.278i)T \)
53 \( 1 + (0.825 + 0.564i)T \)
59 \( 1 + (0.990 - 0.140i)T \)
61 \( 1 + (0.844 + 0.535i)T \)
67 \( 1 + (-0.520 - 0.854i)T \)
71 \( 1 + (0.688 + 0.725i)T \)
73 \( 1 + (0.261 + 0.965i)T \)
79 \( 1 + (-0.0881 - 0.996i)T \)
83 \( 1 + (-0.329 - 0.944i)T \)
89 \( 1 + (0.760 - 0.648i)T \)
97 \( 1 + (0.984 + 0.175i)T \)
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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

−26.72541837054506806770098675308, −26.329634597384778084424255420414, −25.178233800229798557847629180953, −24.28765806468467664719396643032, −23.87319542097720281152253966165, −22.63142684639205522422208286726, −21.17787809286499791020397781895, −19.65236964902422341776788734842, −19.35228664711991321863702405447, −18.56149062919700632981500264889, −17.5737450860052833911571316211, −16.12410147170013079760020506437, −15.502607989825848461983271461471, −14.482134408591537147166030081270, −13.48783293657240666966156746863, −11.95868992648189526296119088094, −11.25670696887990882046286456757, −9.39008961594625282705649862656, −8.680003532939520544113027684807, −7.88214540536444934857029317057, −6.74925935858124565685142294212, −5.83762979301453168051178526279, −3.88958969168131298744136706401, −2.35173308670401064596561558517, −0.83241769914389938835682615100, 0.8253304873490019320973027142, 2.73180366374015369092778348114, 3.77994056414002625738193559490, 4.56913144854978704754876479122, 7.12463251973796024135358814129, 7.86065380300678653681668447663, 8.925291620234563025595976392232, 10.031964923133212891715517830015, 10.7358763388065050110599343259, 11.87987079091847533568413724305, 13.04435242761808853291443841403, 14.268682503684112684511882846474, 15.638639666690365906496230493040, 16.20612139204040001484655764390, 17.340421249395067680404516055206, 18.54666852363513537395733802084, 19.71521027085036194721592758751, 20.31713682961750537198080023736, 20.63078329986960969212541803782, 22.440967437794161506087367801307, 22.72072979628756654295683356122, 24.49484196600640154522377687569, 25.588677771294179560528130417319, 26.41852973294182727189020174933, 27.18975740355881794383815919044

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