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Properties

Label 1-179-179.29-r0-0-0
Degree $1$
Conductor $179$
Sign $0.333 - 0.942i$
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  + (−0.520 − 0.854i)2-s + (−0.825 − 0.564i)3-s + (−0.458 + 0.888i)4-s + (−0.994 + 0.105i)5-s + (−0.0529 + 0.998i)6-s + (−0.635 + 0.772i)7-s + (0.997 − 0.0705i)8-s + (0.362 + 0.932i)9-s + (0.607 + 0.794i)10-s + (0.938 − 0.345i)11-s + (0.880 − 0.474i)12-s + (−0.896 − 0.442i)13-s + (0.990 + 0.140i)14-s + (0.880 + 0.474i)15-s + (−0.579 − 0.815i)16-s + (0.960 + 0.278i)17-s + ⋯
L(s)  = 1  + (−0.520 − 0.854i)2-s + (−0.825 − 0.564i)3-s + (−0.458 + 0.888i)4-s + (−0.994 + 0.105i)5-s + (−0.0529 + 0.998i)6-s + (−0.635 + 0.772i)7-s + (0.997 − 0.0705i)8-s + (0.362 + 0.932i)9-s + (0.607 + 0.794i)10-s + (0.938 − 0.345i)11-s + (0.880 − 0.474i)12-s + (−0.896 − 0.442i)13-s + (0.990 + 0.140i)14-s + (0.880 + 0.474i)15-s + (−0.579 − 0.815i)16-s + (0.960 + 0.278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.333 - 0.942i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.333 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3797554545 - 0.2685655615i\)
\(L(\frac12)\) \(\approx\) \(0.3797554545 - 0.2685655615i\)
\(L(1)\) \(\approx\) \(0.4713104093 - 0.2200818359i\)
\(L(1)\) \(\approx\) \(0.4713104093 - 0.2200818359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.520 - 0.854i)T \)
3 \( 1 + (-0.825 - 0.564i)T \)
5 \( 1 + (-0.994 + 0.105i)T \)
7 \( 1 + (-0.635 + 0.772i)T \)
11 \( 1 + (0.938 - 0.345i)T \)
13 \( 1 + (-0.896 - 0.442i)T \)
17 \( 1 + (0.960 + 0.278i)T \)
19 \( 1 + (0.295 - 0.955i)T \)
23 \( 1 + (-0.999 + 0.0352i)T \)
29 \( 1 + (0.158 + 0.987i)T \)
31 \( 1 + (0.804 + 0.593i)T \)
37 \( 1 + (0.158 - 0.987i)T \)
41 \( 1 + (0.0176 - 0.999i)T \)
43 \( 1 + (0.977 + 0.210i)T \)
47 \( 1 + (0.662 - 0.749i)T \)
53 \( 1 + (0.227 + 0.973i)T \)
59 \( 1 + (0.911 + 0.411i)T \)
61 \( 1 + (-0.123 - 0.992i)T \)
67 \( 1 + (0.997 + 0.0705i)T \)
71 \( 1 + (0.760 + 0.648i)T \)
73 \( 1 + (0.713 - 0.700i)T \)
79 \( 1 + (-0.261 + 0.965i)T \)
83 \( 1 + (0.844 - 0.535i)T \)
89 \( 1 + (-0.520 + 0.854i)T \)
97 \( 1 + (-0.863 + 0.505i)T \)
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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

−27.2776152521647044842782639468, −26.86511600929994151912091511270, −25.84610241533186280399386409250, −24.52011270806666709286313680392, −23.676776927465412893093168884135, −22.80451455254355300633838751935, −22.37973873795564674893962760300, −20.59879826257565642278716512150, −19.58248955567516027797458988945, −18.763560401541475272932980825912, −17.36278576557920868475382351321, −16.6740337145775488789560579604, −16.094473882161591185956514907843, −15.021628732975831933237527822111, −14.10162681175910369921109892354, −12.35827936765239475557017431490, −11.50163836114417719939165173722, −10.03511808220461132682230054199, −9.63000928704462491514740119495, −7.96842827437305488045942609516, −7.01554527786259210340600576194, −6.05152439097651495645137448124, −4.593717857673832541116139461120, −3.85566486565072721963422196618, −0.86938778371190948577078835122, 0.77312876281042342845926779599, 2.52751650618463463027801480868, 3.80318577289788387422341957607, 5.26884411207788585590690857720, 6.806300531237496657860770742430, 7.795547702116719101654288423408, 8.96802619525488314248854657831, 10.2359018177428721989079300124, 11.32239703820809178338594036472, 12.276365861141283141354028665889, 12.459642246431565252381705095924, 14.0893684388702938604479547428, 15.721843474489356872869597898742, 16.60466207786442811075151904578, 17.60245709940621726629716239298, 18.61038319515893273150357288398, 19.4128565333480163865681274857, 19.872378355206149420582532326332, 21.68973231637499026792828857489, 22.24542620922921942994367449898, 23.05103833979575707732489603232, 24.26023488315825266054191439297, 25.264338110007957420200051015636, 26.49572711953814553906986971379, 27.657584092517102032181038667109

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