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Properties

Label 1-173-173.73-r0-0-0
Degree $1$
Conductor $173$
Sign $0.838 - 0.544i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.838 - 0.544i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.838 - 0.544i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.248427424 - 0.6661108600i\)
\(L(\frac12)\) \(\approx\) \(2.248427424 - 0.6661108600i\)
\(L(1)\) \(\approx\) \(2.011814041 - 0.3834550973i\)
\(L(1)\) \(\approx\) \(2.011814041 - 0.3834550973i\)

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

−27.17346841454239125290440017530, −26.46283116737899167270908770238, −25.426217103532941660870837372177, −24.579729438853371285714251779568, −23.9907806272551398546781673052, −22.47965043122823187028488104213, −22.10882817446619344011663346475, −20.88559857763697000049361604088, −19.72454068313844878463332331031, −19.14278601138638654107739684970, −18.01665749088405828413747782988, −16.21198554253580317856210449644, −15.48968822787543674066257456635, −14.53945519639018646742417588981, −13.95401784277108226228662404032, −12.56957019097183389781279383658, −11.95806999481980203540718396326, −10.625731272344895955001837700497, −9.004462638932913093238027542932, −7.87040175633682418463840873067, −6.913382123193342549900926452552, −5.88489076069986619152079627741, −4.178239339058304828221970425783, −3.07459973250687251684044997900, −2.35802313287894213102753824892, 1.690045271421166703416952642703, 3.17890644612534406014580087217, 4.300715711211935334279203646019, 4.82604702713187214094309510218, 6.92693795627363383334334601780, 7.64671565197952537046512904322, 9.256562415692705201432317427429, 10.16585924178741873102784927901, 11.517703521063467776147045337609, 12.67829781796541463494741902949, 13.423142483932318056701830177210, 14.445980582870902351896007002, 15.44107205098585768524615002467, 16.12650853114027860200469488453, 17.33195629968920610112871680210, 19.48305006231737819465922344822, 19.83407554201640490744208736228, 20.4028843757552520786129723873, 21.58425613256915421969158800013, 22.49925565278576554205575962804, 23.6452800611797783025260726380, 24.35741324494514789140487157690, 25.3178876122914514350395364742, 26.360004562595530275035115305854, 27.37976712524460358620485061327

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