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Properties

Label 1-173-173.135-r0-0-0
Degree $1$
Conductor $173$
Sign $0.515 - 0.857i$
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  + (−0.322 − 0.946i)2-s + (0.520 − 0.853i)3-s + (−0.791 + 0.611i)4-s + (0.252 + 0.967i)5-s + (−0.976 − 0.217i)6-s + (−0.181 + 0.983i)7-s + (0.833 + 0.551i)8-s + (−0.457 − 0.889i)9-s + (0.833 − 0.551i)10-s + (0.957 + 0.288i)11-s + (0.109 + 0.994i)12-s + (−0.322 − 0.946i)13-s + (0.989 − 0.145i)14-s + (0.957 + 0.288i)15-s + (0.252 − 0.967i)16-s + (0.905 − 0.424i)17-s + ⋯
L(s)  = 1  + (−0.322 − 0.946i)2-s + (0.520 − 0.853i)3-s + (−0.791 + 0.611i)4-s + (0.252 + 0.967i)5-s + (−0.976 − 0.217i)6-s + (−0.181 + 0.983i)7-s + (0.833 + 0.551i)8-s + (−0.457 − 0.889i)9-s + (0.833 − 0.551i)10-s + (0.957 + 0.288i)11-s + (0.109 + 0.994i)12-s + (−0.322 − 0.946i)13-s + (0.989 − 0.145i)14-s + (0.957 + 0.288i)15-s + (0.252 − 0.967i)16-s + (0.905 − 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.515 - 0.857i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.515 - 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9873163599 - 0.5585403388i\)
\(L(\frac12)\) \(\approx\) \(0.9873163599 - 0.5585403388i\)
\(L(1)\) \(\approx\) \(0.9602378872 - 0.4448186984i\)
\(L(1)\) \(\approx\) \(0.9602378872 - 0.4448186984i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.322 - 0.946i)T \)
3 \( 1 + (0.520 - 0.853i)T \)
5 \( 1 + (0.252 + 0.967i)T \)
7 \( 1 + (-0.181 + 0.983i)T \)
11 \( 1 + (0.957 + 0.288i)T \)
13 \( 1 + (-0.322 - 0.946i)T \)
17 \( 1 + (0.905 - 0.424i)T \)
19 \( 1 + (0.989 + 0.145i)T \)
23 \( 1 + (0.957 - 0.288i)T \)
29 \( 1 + (-0.976 + 0.217i)T \)
31 \( 1 + (0.520 + 0.853i)T \)
37 \( 1 + (0.989 + 0.145i)T \)
41 \( 1 + (-0.181 + 0.983i)T \)
43 \( 1 + (-0.791 - 0.611i)T \)
47 \( 1 + (-0.581 - 0.813i)T \)
53 \( 1 + (0.639 + 0.768i)T \)
59 \( 1 + (-0.934 - 0.357i)T \)
61 \( 1 + (0.905 + 0.424i)T \)
67 \( 1 + (0.520 - 0.853i)T \)
71 \( 1 + (-0.976 + 0.217i)T \)
73 \( 1 + (0.391 - 0.920i)T \)
79 \( 1 + (-0.581 + 0.813i)T \)
83 \( 1 + (-0.872 - 0.489i)T \)
89 \( 1 + (-0.997 - 0.0729i)T \)
97 \( 1 + (-0.872 + 0.489i)T \)
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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

−27.4089333089960364347191679460, −26.61351152989010757041881083044, −25.82981819625640764100523291613, −24.83988678274963694494769094995, −24.08640425115316573554979323114, −22.95348976613054065347105384122, −21.897808030758647019908782123442, −20.8094675363082469413091083550, −19.80592869312447578402054631107, −19.04874011510897674371285611520, −17.215120898570998740629288540182, −16.767889069975824652021940174969, −16.1182146831867095838815017216, −14.78174876490272182144734259483, −13.998401069991036725511333030025, −13.17688506262141104356316597004, −11.36431769364948289966072454206, −9.78705162081878934663885032236, −9.4411451139120314563334339862, −8.31343130869517220554970955191, −7.22001064642632204080143731337, −5.745434092403876918195511207256, −4.59792417595463537445878194123, −3.75819475388644545578519409869, −1.284902056806604643017959591357, 1.42768025063157669371324846318, 2.7558431438576729286847313557, 3.32158890907931271346059094028, 5.46887259308841532847878970561, 6.907437247936159593650976264490, 7.959396668306601451785078694560, 9.18308419526272736853827798826, 9.96577237884440022086971055639, 11.477481793884938362361023030892, 12.179758152605831501508473433085, 13.185934211166763026852696900514, 14.31030332494334865703996682277, 15.04731661029447859417023152676, 16.97682194603404447554585771354, 18.15035503620149910926878508256, 18.4972266991041455668234997773, 19.46699013007216231073001227848, 20.28635007277123938841306347135, 21.48911038940443960894624822206, 22.464794829580379192399746907370, 23.082624674341902783527991083505, 25.00667093281636012552524902385, 25.2398316069571607304135100264, 26.46844223360023379584633082047, 27.33372630217988291421757776392

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