↑

Properties

Label 1-179-179.47-r0-0-0
Degree $1$
Conductor $179$
Sign $0.997 - 0.0696i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.295 + 0.955i)2-s + (0.550 − 0.835i)3-s + (−0.825 + 0.564i)4-s + (0.844 − 0.535i)5-s + (0.960 + 0.278i)6-s + (−0.863 − 0.505i)7-s + (−0.783 − 0.621i)8-s + (−0.394 − 0.918i)9-s + (0.760 + 0.648i)10-s + (0.977 + 0.210i)11-s + (0.0176 + 0.999i)12-s + (0.938 + 0.345i)13-s + (0.227 − 0.973i)14-s + (0.0176 − 0.999i)15-s + (0.362 − 0.932i)16-s + (−0.896 − 0.442i)17-s + ⋯
L(s)  = 1  + (0.295 + 0.955i)2-s + (0.550 − 0.835i)3-s + (−0.825 + 0.564i)4-s + (0.844 − 0.535i)5-s + (0.960 + 0.278i)6-s + (−0.863 − 0.505i)7-s + (−0.783 − 0.621i)8-s + (−0.394 − 0.918i)9-s + (0.760 + 0.648i)10-s + (0.977 + 0.210i)11-s + (0.0176 + 0.999i)12-s + (0.938 + 0.345i)13-s + (0.227 − 0.973i)14-s + (0.0176 − 0.999i)15-s + (0.362 − 0.932i)16-s + (−0.896 − 0.442i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.997 - 0.0696i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.997 - 0.0696i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.534029604 - 0.05351832819i\)
\(L(\frac12)\) \(\approx\) \(1.534029604 - 0.05351832819i\)
\(L(1)\) \(\approx\) \(1.386385667 + 0.09832921128i\)
\(L(1)\) \(\approx\) \(1.386385667 + 0.09832921128i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.295 + 0.955i)T \)
3 \( 1 + (0.550 - 0.835i)T \)
5 \( 1 + (0.844 - 0.535i)T \)
7 \( 1 + (-0.863 - 0.505i)T \)
11 \( 1 + (0.977 + 0.210i)T \)
13 \( 1 + (0.938 + 0.345i)T \)
17 \( 1 + (-0.896 - 0.442i)T \)
19 \( 1 + (0.880 - 0.474i)T \)
23 \( 1 + (-0.329 + 0.944i)T \)
29 \( 1 + (0.662 - 0.749i)T \)
31 \( 1 + (-0.969 - 0.244i)T \)
37 \( 1 + (0.662 + 0.749i)T \)
41 \( 1 + (-0.579 + 0.815i)T \)
43 \( 1 + (0.427 + 0.904i)T \)
47 \( 1 + (-0.192 + 0.981i)T \)
53 \( 1 + (-0.984 + 0.175i)T \)
59 \( 1 + (-0.635 + 0.772i)T \)
61 \( 1 + (-0.925 - 0.378i)T \)
67 \( 1 + (-0.783 + 0.621i)T \)
71 \( 1 + (0.911 - 0.411i)T \)
73 \( 1 + (-0.458 - 0.888i)T \)
79 \( 1 + (0.158 + 0.987i)T \)
83 \( 1 + (0.607 - 0.794i)T \)
89 \( 1 + (0.295 - 0.955i)T \)
97 \( 1 + (-0.949 - 0.312i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.419694551209650289712755147183, −26.51058646185883191289281165839, −25.61628462218417366122199711000, −24.70739721017880813620810901114, −22.95793805012796449090744737028, −22.0410339270045402873264219453, −21.88631855321300206923039795040, −20.59683509624253354551123031950, −19.865199975354678394497951116592, −18.82992346516385468924135566584, −17.93203759262787927960625135918, −16.50744368471352952475128432482, −15.312541163151244070280654837255, −14.28976878884922046899103730069, −13.6146432436085917055491271934, −12.491724907007783911065451346507, −11.05139465248623571720284918482, −10.306124231479706894837356931975, −9.30124498263560256644888940378, −8.7062633127680101115964597147, −6.39986232884405292487736153531, −5.44076287158374960574422822756, −3.86471721506595270604502446797, −3.093491880558752275830961205801, −1.902901177648156552429418487853, 1.24753842585830255421879040932, 3.10667135383134594395796209331, 4.38813792770601410467144283797, 6.090886843688459622893800049070, 6.59003759731072745726744922274, 7.76197934911576507593665487996, 9.15836005503890877608793058337, 9.457721517082646713758404052353, 11.73995765190007455899696723909, 12.99611494068911487698583924719, 13.54741911867805251390109323449, 14.175120766666085059784682419977, 15.58582859023820032957118299028, 16.59508463779320644920730000008, 17.55646857227957094227505376985, 18.28025232077350087394622032701, 19.61443727221037502568237948631, 20.489467667731607361168645717629, 21.79748525793336716606953192010, 22.7599598392573051708470500606, 23.7727468314810695772917164396, 24.57706999347971299342898733103, 25.4260221501531382960831498072, 25.89298886479291722029982877272, 26.93221182679668773904952257126

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