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Properties

Label 1-173-173.51-r0-0-0
Degree $1$
Conductor $173$
Sign $0.168 + 0.985i$
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  + (−0.872 − 0.489i)2-s + (−0.322 − 0.946i)3-s + (0.520 + 0.853i)4-s + (−0.457 − 0.889i)5-s + (−0.181 + 0.983i)6-s + (0.109 + 0.994i)7-s + (−0.0365 − 0.999i)8-s + (−0.791 + 0.611i)9-s + (−0.0365 + 0.999i)10-s + (−0.694 + 0.719i)11-s + (0.639 − 0.768i)12-s + (−0.872 − 0.489i)13-s + (0.391 − 0.920i)14-s + (−0.694 + 0.719i)15-s + (−0.457 + 0.889i)16-s + (−0.934 + 0.357i)17-s + ⋯
L(s)  = 1  + (−0.872 − 0.489i)2-s + (−0.322 − 0.946i)3-s + (0.520 + 0.853i)4-s + (−0.457 − 0.889i)5-s + (−0.181 + 0.983i)6-s + (0.109 + 0.994i)7-s + (−0.0365 − 0.999i)8-s + (−0.791 + 0.611i)9-s + (−0.0365 + 0.999i)10-s + (−0.694 + 0.719i)11-s + (0.639 − 0.768i)12-s + (−0.872 − 0.489i)13-s + (0.391 − 0.920i)14-s + (−0.694 + 0.719i)15-s + (−0.457 + 0.889i)16-s + (−0.934 + 0.357i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.168 + 0.985i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.168 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1066002789 + 0.08994092913i\)
\(L(\frac12)\) \(\approx\) \(0.1066002789 + 0.08994092913i\)
\(L(1)\) \(\approx\) \(0.3970325058 - 0.1447168962i\)
\(L(1)\) \(\approx\) \(0.3970325058 - 0.1447168962i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.872 - 0.489i)T \)
3 \( 1 + (-0.322 - 0.946i)T \)
5 \( 1 + (-0.457 - 0.889i)T \)
7 \( 1 + (0.109 + 0.994i)T \)
11 \( 1 + (-0.694 + 0.719i)T \)
13 \( 1 + (-0.872 - 0.489i)T \)
17 \( 1 + (-0.934 + 0.357i)T \)
19 \( 1 + (0.391 + 0.920i)T \)
23 \( 1 + (-0.694 - 0.719i)T \)
29 \( 1 + (-0.181 - 0.983i)T \)
31 \( 1 + (-0.322 + 0.946i)T \)
37 \( 1 + (0.391 + 0.920i)T \)
41 \( 1 + (0.109 + 0.994i)T \)
43 \( 1 + (0.520 - 0.853i)T \)
47 \( 1 + (0.252 + 0.967i)T \)
53 \( 1 + (0.744 + 0.667i)T \)
59 \( 1 + (-0.976 + 0.217i)T \)
61 \( 1 + (-0.934 - 0.357i)T \)
67 \( 1 + (-0.322 - 0.946i)T \)
71 \( 1 + (-0.181 - 0.983i)T \)
73 \( 1 + (-0.997 - 0.0729i)T \)
79 \( 1 + (0.252 - 0.967i)T \)
83 \( 1 + (-0.581 - 0.813i)T \)
89 \( 1 + (0.833 + 0.551i)T \)
97 \( 1 + (-0.581 + 0.813i)T \)
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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

−27.05352116292719488772517444043, −26.34637714449633202946744352025, −26.08691309324611677725106951040, −24.218665159633138125839439948433, −23.601911030372850106514350536341, −22.5217281486612611743616049897, −21.53058194587563251485989907993, −20.16378892881522272339722792066, −19.57141980322015907048906463691, −18.23414787840182449641828405778, −17.43364245258032151946428048443, −16.36366041542357148146614190232, −15.68718731155794926594664795027, −14.69472856999048810116996987089, −13.79284052203213456848593001898, −11.49915549488825553598036553806, −10.959494249482091708310409925134, −10.101686518765510625586511879659, −9.0671385211311893656863736814, −7.648587460473029141516534648943, −6.84092072252912059254285806333, −5.48117883778397761347747138499, −4.16039062986456380244577100111, −2.660028315176478317084167673862, −0.14821815346208261444795913912, 1.66218460649305053220576303635, 2.67194317892073381656898303733, 4.698274136759204049672659321451, 6.075703245460162086671571105813, 7.60579684468508244617239103002, 8.16830227117587700611843574903, 9.27201282877596475888988743070, 10.60712616237682764802703966109, 12.04385999168008796710060999511, 12.25749075826828533467238153365, 13.21396225896863690816929013992, 15.12647621632419633855785177209, 16.15999413982627625672386759022, 17.23672170094875575926139716234, 18.02494770637336410427372637609, 18.84531660307234152995673207027, 19.85446399456326714693574694462, 20.51569245051524936250791718329, 21.821440048131432958940379451246, 22.87562515038337253237998405266, 24.28118047691601193721862477757, 24.76260629641641885716875102254, 25.66309413596822497250868211957, 26.99359779694579940369439464023, 27.984851586666454194896086135794

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