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Properties

Label 1-173-173.49-r0-0-0
Degree $1$
Conductor $173$
Sign $0.905 + 0.425i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.905 + 0.425i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.905 + 0.425i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.121641230 + 0.4735989040i\)
\(L(\frac12)\) \(\approx\) \(2.121641230 + 0.4735989040i\)
\(L(1)\) \(\approx\) \(1.874022707 + 0.3236198238i\)
\(L(1)\) \(\approx\) \(1.874022707 + 0.3236198238i\)

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

−27.565765597896438305613318790306, −26.6022407867904918458060322497, −25.329612432933420307490598015957, −24.44822196574567576976200354307, −23.615828794669575323438470476954, −22.073652364055732254031274604811, −21.37832390439732643038374757423, −21.11086989011703348510736086416, −19.99205161737305779522312408535, −18.97145132661310471893521032311, −17.58740755093326374785245531259, −16.42128426383714134755047767412, −15.20424212112151469639986504464, −14.48009397590750147458123173110, −13.56861416326682796181846396833, −12.569993012768463631267387907520, −11.129736019855378601939506500907, −10.4374827704622450336292576724, −9.269711257222317411850540049285, −8.34949657941427168932841341861, −6.251489716332019610791608254575, −4.98207574666722954453781965912, −4.56124132430803622280128512826, −2.78848556309286202624984777449, −1.90858283590969110487136056504, 2.04140916820326217871728423699, 2.88052089550982103108840283111, 4.69580935773726750227730256717, 5.771970355163241518990760600931, 7.06429635911720400035012208828, 7.62744862966540321873235674155, 8.8815244423324934149966128237, 10.52426405922100176476456461147, 11.818016747495404244894469780879, 13.03564159224699400771402088392, 13.57577512914140748006197895243, 14.681063359256865104828160589224, 15.1387676878402390175537783121, 17.01142629120137767135063184719, 17.70350500501950014602375424790, 18.412288674800050186130164615970, 20.08685644190019537149237296115, 20.86002338408623152337079441407, 21.811420683031682999258097059914, 23.0428761563293805242606241560, 23.71630105530681387537717214527, 24.83153375359627371121746962511, 25.2749969341779633052859758091, 26.25807431022456150618503004345, 27.16104144092001504166549796586

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