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Properties

Label 1-173-173.40-r0-0-0
Degree $1$
Conductor $173$
Sign $0.992 - 0.124i$
Analytic cond. $0.803408$
Root an. cond. $0.803408$
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  + (0.0365 + 0.999i)2-s + (−0.833 − 0.551i)3-s + (−0.997 + 0.0729i)4-s + (−0.989 − 0.145i)5-s + (0.520 − 0.853i)6-s + (0.322 + 0.946i)7-s + (−0.109 − 0.994i)8-s + (0.391 + 0.920i)9-s + (0.109 − 0.994i)10-s + (−0.744 − 0.667i)11-s + (0.872 + 0.489i)12-s + (−0.0365 − 0.999i)13-s + (−0.934 + 0.357i)14-s + (0.744 + 0.667i)15-s + (0.989 − 0.145i)16-s + (0.457 − 0.889i)17-s + ⋯
L(s)  = 1  + (0.0365 + 0.999i)2-s + (−0.833 − 0.551i)3-s + (−0.997 + 0.0729i)4-s + (−0.989 − 0.145i)5-s + (0.520 − 0.853i)6-s + (0.322 + 0.946i)7-s + (−0.109 − 0.994i)8-s + (0.391 + 0.920i)9-s + (0.109 − 0.994i)10-s + (−0.744 − 0.667i)11-s + (0.872 + 0.489i)12-s + (−0.0365 − 0.999i)13-s + (−0.934 + 0.357i)14-s + (0.744 + 0.667i)15-s + (0.989 − 0.145i)16-s + (0.457 − 0.889i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.992 - 0.124i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.992 - 0.124i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5754850924 - 0.03594910147i\)
\(L(\frac12)\) \(\approx\) \(0.5754850924 - 0.03594910147i\)
\(L(1)\) \(\approx\) \(0.6334472531 + 0.1276358971i\)
\(L(1)\) \(\approx\) \(0.6334472531 + 0.1276358971i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.0365 + 0.999i)T \)
3 \( 1 + (-0.833 - 0.551i)T \)
5 \( 1 + (-0.989 - 0.145i)T \)
7 \( 1 + (0.322 + 0.946i)T \)
11 \( 1 + (-0.744 - 0.667i)T \)
13 \( 1 + (-0.0365 - 0.999i)T \)
17 \( 1 + (0.457 - 0.889i)T \)
19 \( 1 + (0.934 + 0.357i)T \)
23 \( 1 + (0.744 - 0.667i)T \)
29 \( 1 + (0.520 + 0.853i)T \)
31 \( 1 + (0.833 - 0.551i)T \)
37 \( 1 + (-0.934 - 0.357i)T \)
41 \( 1 + (-0.322 - 0.946i)T \)
43 \( 1 + (-0.997 - 0.0729i)T \)
47 \( 1 + (-0.694 - 0.719i)T \)
53 \( 1 + (0.581 - 0.813i)T \)
59 \( 1 + (0.791 - 0.611i)T \)
61 \( 1 + (0.457 + 0.889i)T \)
67 \( 1 + (0.833 + 0.551i)T \)
71 \( 1 + (-0.520 - 0.853i)T \)
73 \( 1 + (-0.976 - 0.217i)T \)
79 \( 1 + (0.694 - 0.719i)T \)
83 \( 1 + (0.957 - 0.288i)T \)
89 \( 1 + (-0.181 + 0.983i)T \)
97 \( 1 + (-0.957 - 0.288i)T \)
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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

−27.71209031922923493091960424909, −26.61468427415160242815863134495, −26.41143911660696597627000592446, −23.97143553580026381862821572856, −23.33934750204874720235259975758, −22.82824492735034504353348875820, −21.5471332065069120902358701271, −20.829185426390540958949053316158, −19.85185849466085432315685876403, −18.86578731011077637274502761969, −17.7405041279498064400385922132, −16.9095038659849254127746962566, −15.692351112275916123556089385, −14.62033254720591593749080840519, −13.316995538375474229861542822935, −12.076134306762948130657835701134, −11.43045574567500138106017794074, −10.51409751605751422840654793898, −9.710606085313824971330735080725, −8.16001004922813296902572935276, −6.91581378825180483929566761936, −5.03778689064651761517648264977, −4.32680245718775722792507925340, −3.30063427743222108986614649752, −1.20456997619996411330394971614, 0.64116589664378927297979118845, 3.14980596845113634656498302537, 5.0617640588666368639518929382, 5.42582173123051476922182267291, 6.88798534074036174665519416348, 7.884897816386554871196927280752, 8.586890067754601747318976064215, 10.3260023824561908248199893105, 11.67764080140213812191754963731, 12.461097113838881068324626837287, 13.493426912416658224218367346982, 14.86702511353719329133213166913, 15.88791105220557852936116581211, 16.374043035008121825042023193064, 17.74678079094346982779002430626, 18.483053823539290765021109062039, 19.136299248645746444113943241424, 20.84333920605476940935542495850, 22.22821171228825848100147513935, 22.839377470721441355366652833605, 23.71967031488469549301309297449, 24.58536317922030156134094064160, 25.0868246155789241147537630044, 26.65749225207212094280820199418, 27.47116233059369681636636846335

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