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Properties

Label 1-179-179.117-r0-0-0
Degree $1$
Conductor $179$
Sign $0.572 - 0.819i$
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  + (0.607 − 0.794i)2-s + (0.158 + 0.987i)3-s + (−0.261 − 0.965i)4-s + (0.550 − 0.835i)5-s + (0.880 + 0.474i)6-s + (0.990 + 0.140i)7-s + (−0.925 − 0.378i)8-s + (−0.949 + 0.312i)9-s + (−0.329 − 0.944i)10-s + (0.362 − 0.932i)11-s + (0.911 − 0.411i)12-s + (−0.579 + 0.815i)13-s + (0.713 − 0.700i)14-s + (0.911 + 0.411i)15-s + (−0.863 + 0.505i)16-s + (0.0176 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.607 − 0.794i)2-s + (0.158 + 0.987i)3-s + (−0.261 − 0.965i)4-s + (0.550 − 0.835i)5-s + (0.880 + 0.474i)6-s + (0.990 + 0.140i)7-s + (−0.925 − 0.378i)8-s + (−0.949 + 0.312i)9-s + (−0.329 − 0.944i)10-s + (0.362 − 0.932i)11-s + (0.911 − 0.411i)12-s + (−0.579 + 0.815i)13-s + (0.713 − 0.700i)14-s + (0.911 + 0.411i)15-s + (−0.863 + 0.505i)16-s + (0.0176 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.572 - 0.819i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.572 - 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.564477481 - 0.8157360914i\)
\(L(\frac12)\) \(\approx\) \(1.564477481 - 0.8157360914i\)
\(L(1)\) \(\approx\) \(1.483731660 - 0.5144612627i\)
\(L(1)\) \(\approx\) \(1.483731660 - 0.5144612627i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.607 - 0.794i)T \)
3 \( 1 + (0.158 + 0.987i)T \)
5 \( 1 + (0.550 - 0.835i)T \)
7 \( 1 + (0.990 + 0.140i)T \)
11 \( 1 + (0.362 - 0.932i)T \)
13 \( 1 + (-0.579 + 0.815i)T \)
17 \( 1 + (0.0176 - 0.999i)T \)
19 \( 1 + (0.760 + 0.648i)T \)
23 \( 1 + (-0.192 + 0.981i)T \)
29 \( 1 + (0.0881 - 0.996i)T \)
31 \( 1 + (0.938 - 0.345i)T \)
37 \( 1 + (0.0881 + 0.996i)T \)
41 \( 1 + (-0.635 + 0.772i)T \)
43 \( 1 + (-0.394 + 0.918i)T \)
47 \( 1 + (-0.0529 - 0.998i)T \)
53 \( 1 + (-0.458 + 0.888i)T \)
59 \( 1 + (-0.688 - 0.725i)T \)
61 \( 1 + (-0.994 + 0.105i)T \)
67 \( 1 + (-0.925 + 0.378i)T \)
71 \( 1 + (-0.737 + 0.675i)T \)
73 \( 1 + (0.427 + 0.904i)T \)
79 \( 1 + (-0.783 + 0.621i)T \)
83 \( 1 + (-0.999 + 0.0352i)T \)
89 \( 1 + (0.607 + 0.794i)T \)
97 \( 1 + (0.227 + 0.973i)T \)
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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

−27.16686046508091379276088788559, −26.178539252094207259668353934502, −25.38644726512719003667515057633, −24.604968283116233693224040845668, −23.8451359550964926213800855988, −22.77429631887073408670327957111, −22.10977721488721898093433979509, −20.86028258808741030318947315241, −19.80415650926107275916514346264, −18.28692471663193003718046187009, −17.681559689611039921549944432381, −17.125391884094011408465450323846, −15.23955582321086296739596149257, −14.560036601716867671167832176017, −13.89951241003828086052689507967, −12.75418190903155772866804693103, −11.90681935591707128530881342070, −10.539483790264378279444461730415, −8.86376708517901546406677431980, −7.68153207638844535833422161531, −7.03585274966337532134404913928, −5.97195051724191254218097365730, −4.80698470334111607021151919765, −3.11496101649955177936353751994, −1.91952451911252892421074030756, 1.435680211148726273504111260734, 2.855095035269609192811576811703, 4.28761106108229912934886258515, 5.03042168102773721192050477945, 5.94373867491985119095728085289, 8.24498264907160445626018334190, 9.346272597418472948988073457429, 9.96845506028099260293952228725, 11.52748770285699270481897834453, 11.786221370967997322322460750632, 13.72004876866951759042500490953, 13.99727357731577563914637851126, 15.210811599422464576189262024527, 16.360905780558277804300820711471, 17.34481780998779369063830840990, 18.670784378874116946009809390236, 19.92445290816593538067386908087, 20.67737698565395141812449872341, 21.44854752667722733540022883576, 21.90131766357632947534201867833, 23.18932398289893513599282120848, 24.39732070210008182746461402714, 24.92268279285161675518429428972, 26.69237813018315218201583263235, 27.337911914429402778084735896383

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