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Properties

Label 1-173-173.42-r1-0-0
Degree $1$
Conductor $173$
Sign $0.444 - 0.895i$
Analytic cond. $18.5914$
Root an. cond. $18.5914$
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  + (−0.217 − 0.976i)2-s + (−0.357 + 0.934i)3-s + (−0.905 + 0.424i)4-s + (0.768 − 0.639i)5-s + (0.989 + 0.145i)6-s + (0.920 − 0.391i)7-s + (0.611 + 0.791i)8-s + (−0.744 − 0.667i)9-s + (−0.791 − 0.611i)10-s + (−0.946 + 0.322i)11-s + (−0.0729 − 0.997i)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 + (0.288 + 0.957i)17-s + ⋯
L(s)  = 1  + (−0.217 − 0.976i)2-s + (−0.357 + 0.934i)3-s + (−0.905 + 0.424i)4-s + (0.768 − 0.639i)5-s + (0.989 + 0.145i)6-s + (0.920 − 0.391i)7-s + (0.611 + 0.791i)8-s + (−0.744 − 0.667i)9-s + (−0.791 − 0.611i)10-s + (−0.946 + 0.322i)11-s + (−0.0729 − 0.997i)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 + (0.288 + 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(18.5914\)
Root analytic conductor: \(18.5914\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (1:\ ),\ 0.444 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.333855397 - 0.8274333327i\)
\(L(\frac12)\) \(\approx\) \(1.333855397 - 0.8274333327i\)
\(L(1)\) \(\approx\) \(0.9570015239 - 0.3305130183i\)
\(L(1)\) \(\approx\) \(0.9570015239 - 0.3305130183i\)

Euler product

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

−27.30249924810849075188904029273, −26.19372547838059963555045118517, −25.299855046770639199613300684076, −24.71483028663160850113990152774, −23.61888167400703785428223677689, −23.02542081881808132195598635412, −21.85322005002493452821307515769, −20.78310484513298887514142120371, −18.87469729400315917195716176239, −18.42758028925115209501744241313, −17.80561514826507311583965583158, −16.7967291238432552500748382114, −15.63456046784228680502507886363, −14.27468398946804804424117265824, −13.84278750722951485020762013120, −12.69750600167230092181791112543, −11.20049487148460178980948975076, −10.23047404470563803853889644210, −8.59779621301649741211712836487, −7.89752452900523956613102452001, −6.612636116996485303251371192401, −5.89419325825008472291982797326, −4.86190022447456143504559642914, −2.55509921460180616496276188142, −1.10337384559567004881728035405, 0.81348562977338988847893539387, 2.23508423955197714324906475350, 3.90616984230774321792223340532, 4.828892871149051683388820808169, 5.792867843313732405817796748690, 8.11292695015521647027687636833, 8.90703899456751139493850405953, 10.20169161242942950996955320733, 10.64782120438492617899662126794, 11.79187691495939892999689716445, 13.01933599252784057425212831551, 13.90424383374684778902974508869, 15.218666847927898753280584417557, 16.52632364861714405626565779696, 17.60271686721059701530367079796, 17.866915842061686513442693609582, 19.5841571938296641435510055521, 20.73590587576106329417917848332, 21.08996097953485343814950961531, 21.74425764434683782543949085630, 23.1547637393744200013721294257, 23.800311600272195370460982564382, 25.63152848913142848405109014280, 26.248300320726367223089774724245, 27.45828104122974682096039167303

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