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Properties

Label 1-173-173.60-r0-0-0
Degree $1$
Conductor $173$
Sign $0.833 + 0.552i$
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  + (−0.791 + 0.611i)2-s + (−0.457 − 0.889i)3-s + (0.252 − 0.967i)4-s + (−0.872 + 0.489i)5-s + (0.905 + 0.424i)6-s + (−0.934 − 0.357i)7-s + (0.391 + 0.920i)8-s + (−0.581 + 0.813i)9-s + (0.391 − 0.920i)10-s + (0.833 + 0.551i)11-s + (−0.976 + 0.217i)12-s + (−0.791 + 0.611i)13-s + (0.957 − 0.288i)14-s + (0.833 + 0.551i)15-s + (−0.872 − 0.489i)16-s + (0.639 − 0.768i)17-s + ⋯
L(s)  = 1  + (−0.791 + 0.611i)2-s + (−0.457 − 0.889i)3-s + (0.252 − 0.967i)4-s + (−0.872 + 0.489i)5-s + (0.905 + 0.424i)6-s + (−0.934 − 0.357i)7-s + (0.391 + 0.920i)8-s + (−0.581 + 0.813i)9-s + (0.391 − 0.920i)10-s + (0.833 + 0.551i)11-s + (−0.976 + 0.217i)12-s + (−0.791 + 0.611i)13-s + (0.957 − 0.288i)14-s + (0.833 + 0.551i)15-s + (−0.872 − 0.489i)16-s + (0.639 − 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.833 + 0.552i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.833 + 0.552i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4510536589 + 0.1359376079i\)
\(L(\frac12)\) \(\approx\) \(0.4510536589 + 0.1359376079i\)
\(L(1)\) \(\approx\) \(0.5229837296 + 0.05814530626i\)
\(L(1)\) \(\approx\) \(0.5229837296 + 0.05814530626i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.791 + 0.611i)T \)
3 \( 1 + (-0.457 - 0.889i)T \)
5 \( 1 + (-0.872 + 0.489i)T \)
7 \( 1 + (-0.934 - 0.357i)T \)
11 \( 1 + (0.833 + 0.551i)T \)
13 \( 1 + (-0.791 + 0.611i)T \)
17 \( 1 + (0.639 - 0.768i)T \)
19 \( 1 + (0.957 + 0.288i)T \)
23 \( 1 + (0.833 - 0.551i)T \)
29 \( 1 + (0.905 - 0.424i)T \)
31 \( 1 + (-0.457 + 0.889i)T \)
37 \( 1 + (0.957 + 0.288i)T \)
41 \( 1 + (-0.934 - 0.357i)T \)
43 \( 1 + (0.252 + 0.967i)T \)
47 \( 1 + (-0.322 + 0.946i)T \)
53 \( 1 + (-0.181 + 0.983i)T \)
59 \( 1 + (0.744 + 0.667i)T \)
61 \( 1 + (0.639 + 0.768i)T \)
67 \( 1 + (-0.457 - 0.889i)T \)
71 \( 1 + (0.905 - 0.424i)T \)
73 \( 1 + (-0.694 - 0.719i)T \)
79 \( 1 + (-0.322 - 0.946i)T \)
83 \( 1 + (0.520 + 0.853i)T \)
89 \( 1 + (0.989 + 0.145i)T \)
97 \( 1 + (0.520 - 0.853i)T \)
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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

−27.30807143646108630968161309883, −26.96039113295315314546754323195, −25.76807098769480358020923291349, −24.74455403220897541450894063716, −23.30918377066175009068797670826, −22.21908118793494943678890009465, −21.69794944816718081274399580521, −20.35708911583609731404130617310, −19.70482803082383372791761288814, −18.866686391537232510091999120, −17.38790011988567626768694237626, −16.626497713329680751554803733928, −15.92134394386726254485734917151, −14.92837204523798005171466126782, −12.94603099792094105167149145778, −12.01595024880005273119364865296, −11.33591061240587719192108800836, −10.056728585590623605654715010493, −9.2915690817809726001750647067, −8.31862251991737933650241579571, −6.902021286808279908367431081968, −5.35468380988672928607503202716, −3.82359348251320294520749819150, −3.12811844056199472662925201461, −0.71474808438080030294757135192, 1.00953306131604260437590526823, 2.83186215718747273358376999751, 4.76782150349852628800816838512, 6.365768669159073284731207757233, 7.05966150956405882325337207297, 7.697064355215144745579990794450, 9.20722299372755366266203598124, 10.294788063888060905504446029691, 11.562813526561273811146457730924, 12.295951265540949640173099772514, 13.91228893036755317925211728132, 14.74515037453156305827335307963, 16.16010243886229603545186749216, 16.71082414995706069910768852142, 17.84733898462864703738477506165, 18.82887671863904797171599726544, 19.44579440249279203609543229366, 20.15132949154917687781067745671, 22.41275169926912971612946423186, 22.9556465147365160527224620388, 23.761921756290566782358388786890, 24.84192100044222193820585350531, 25.52557259678450288925088981488, 26.74922568797321376652871497781, 27.344508975397604043209175265801

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