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Properties

Label 1-173-173.100-r0-0-0
Degree $1$
Conductor $173$
Sign $0.0557 + 0.998i$
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  + (−0.976 + 0.217i)2-s + (−0.934 − 0.357i)3-s + (0.905 − 0.424i)4-s + (0.639 + 0.768i)5-s + (0.989 + 0.145i)6-s + (0.391 + 0.920i)7-s + (−0.791 + 0.611i)8-s + (0.744 + 0.667i)9-s + (−0.791 − 0.611i)10-s + (−0.322 − 0.946i)11-s + (−0.997 + 0.0729i)12-s + (−0.976 + 0.217i)13-s + (−0.581 − 0.813i)14-s + (−0.322 − 0.946i)15-s + (0.639 − 0.768i)16-s + (0.957 − 0.288i)17-s + ⋯
L(s)  = 1  + (−0.976 + 0.217i)2-s + (−0.934 − 0.357i)3-s + (0.905 − 0.424i)4-s + (0.639 + 0.768i)5-s + (0.989 + 0.145i)6-s + (0.391 + 0.920i)7-s + (−0.791 + 0.611i)8-s + (0.744 + 0.667i)9-s + (−0.791 − 0.611i)10-s + (−0.322 − 0.946i)11-s + (−0.997 + 0.0729i)12-s + (−0.976 + 0.217i)13-s + (−0.581 − 0.813i)14-s + (−0.322 − 0.946i)15-s + (0.639 − 0.768i)16-s + (0.957 − 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0557 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0557 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.0557 + 0.998i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.0557 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3908672347 + 0.3696381725i\)
\(L(\frac12)\) \(\approx\) \(0.3908672347 + 0.3696381725i\)
\(L(1)\) \(\approx\) \(0.5518716499 + 0.1784555879i\)
\(L(1)\) \(\approx\) \(0.5518716499 + 0.1784555879i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.976 + 0.217i)T \)
3 \( 1 + (-0.934 - 0.357i)T \)
5 \( 1 + (0.639 + 0.768i)T \)
7 \( 1 + (0.391 + 0.920i)T \)
11 \( 1 + (-0.322 - 0.946i)T \)
13 \( 1 + (-0.976 + 0.217i)T \)
17 \( 1 + (0.957 - 0.288i)T \)
19 \( 1 + (-0.581 + 0.813i)T \)
23 \( 1 + (-0.322 + 0.946i)T \)
29 \( 1 + (0.989 - 0.145i)T \)
31 \( 1 + (-0.934 + 0.357i)T \)
37 \( 1 + (-0.581 + 0.813i)T \)
41 \( 1 + (0.391 + 0.920i)T \)
43 \( 1 + (0.905 + 0.424i)T \)
47 \( 1 + (0.109 - 0.994i)T \)
53 \( 1 + (0.833 + 0.551i)T \)
59 \( 1 + (-0.694 + 0.719i)T \)
61 \( 1 + (0.957 + 0.288i)T \)
67 \( 1 + (-0.934 - 0.357i)T \)
71 \( 1 + (0.989 - 0.145i)T \)
73 \( 1 + (0.252 + 0.967i)T \)
79 \( 1 + (0.109 + 0.994i)T \)
83 \( 1 + (-0.181 - 0.983i)T \)
89 \( 1 + (-0.457 - 0.889i)T \)
97 \( 1 + (-0.181 + 0.983i)T \)
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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

−27.568769443901611267811286596410, −26.46750163956553660333572439364, −25.5607353714960569906109375887, −24.34178101889540776898740243044, −23.63515574958741631402858189481, −22.25769455702301607435039082052, −21.13731388447200504783049189522, −20.59046934499605194976420734254, −19.53730321075931283348056923858, −18.008878454852413917398361846777, −17.39213037423583131224804285566, −16.84411407822166670096243907603, −15.878269110279348692463653629348, −14.5653994699148224315529471843, −12.74021899091470441479205773730, −12.21129509738453340317490304293, −10.71255280939620689991633722281, −10.168103099642686522321534535617, −9.22029159980940113490603477488, −7.74566715991891461806443265985, −6.74147735612319023824098821702, −5.32383381035636498035298005962, −4.225779300710405340337991858671, −2.10442896150198025855864583873, −0.68825304066052211351376396232, 1.57404975391318759419119683204, 2.75297121919588442209370934255, 5.40657661994788373245669422775, 5.96574583174646384299154684622, 7.13557362833402933101576682728, 8.17778560774327419194591243923, 9.637817984062590775533060380610, 10.49929529469666634526248524506, 11.48696375358468696877202094833, 12.31168375629816441136847683742, 14.00827291274848891266541267558, 15.048615555149870353969242063194, 16.24007407355263795317141022938, 17.11429106024988206793954117984, 18.04000767473263943947660921296, 18.66258104704595268366466390753, 19.36711301237849528831079064391, 21.32705832132662570178232316330, 21.64109759915581345674253244906, 23.05014136290725840518785630611, 24.119949796893256622376113809086, 24.92993048411145988557745213023, 25.71885411876490899697771190862, 27.05205406897067269330947090563, 27.56240593261829732916056308899

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