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Properties

Label 1-173-173.27-r1-0-0
Degree $1$
Conductor $173$
Sign $0.777 + 0.629i$
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  + (−0.983 + 0.181i)2-s + (−0.217 − 0.976i)3-s + (0.934 − 0.357i)4-s + (−0.667 + 0.744i)5-s + (0.391 + 0.920i)6-s + (−0.0729 − 0.997i)7-s + (−0.853 + 0.520i)8-s + (−0.905 + 0.424i)9-s + (0.520 − 0.853i)10-s + (0.489 − 0.872i)11-s + (−0.551 − 0.833i)12-s + (0.181 + 0.983i)13-s + (0.252 + 0.967i)14-s + (0.872 + 0.489i)15-s + (0.744 − 0.667i)16-s + (−0.719 + 0.694i)17-s + ⋯
L(s)  = 1  + (−0.983 + 0.181i)2-s + (−0.217 − 0.976i)3-s + (0.934 − 0.357i)4-s + (−0.667 + 0.744i)5-s + (0.391 + 0.920i)6-s + (−0.0729 − 0.997i)7-s + (−0.853 + 0.520i)8-s + (−0.905 + 0.424i)9-s + (0.520 − 0.853i)10-s + (0.489 − 0.872i)11-s + (−0.551 − 0.833i)12-s + (0.181 + 0.983i)13-s + (0.252 + 0.967i)14-s + (0.872 + 0.489i)15-s + (0.744 − 0.667i)16-s + (−0.719 + 0.694i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.777 + 0.629i$
Analytic conductor: \(18.5914\)
Root analytic conductor: \(18.5914\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (1:\ ),\ 0.777 + 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5310414681 + 0.1880248803i\)
\(L(\frac12)\) \(\approx\) \(0.5310414681 + 0.1880248803i\)
\(L(1)\) \(\approx\) \(0.5311725422 - 0.06138136522i\)
\(L(1)\) \(\approx\) \(0.5311725422 - 0.06138136522i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.983 + 0.181i)T \)
3 \( 1 + (-0.217 - 0.976i)T \)
5 \( 1 + (-0.667 + 0.744i)T \)
7 \( 1 + (-0.0729 - 0.997i)T \)
11 \( 1 + (0.489 - 0.872i)T \)
13 \( 1 + (0.181 + 0.983i)T \)
17 \( 1 + (-0.719 + 0.694i)T \)
19 \( 1 + (-0.967 - 0.252i)T \)
23 \( 1 + (-0.872 + 0.489i)T \)
29 \( 1 + (0.391 - 0.920i)T \)
31 \( 1 + (0.976 + 0.217i)T \)
37 \( 1 + (-0.252 + 0.967i)T \)
41 \( 1 + (0.997 - 0.0729i)T \)
43 \( 1 + (-0.934 - 0.357i)T \)
47 \( 1 + (0.639 + 0.768i)T \)
53 \( 1 + (0.999 - 0.0365i)T \)
59 \( 1 + (0.145 + 0.989i)T \)
61 \( 1 + (0.719 + 0.694i)T \)
67 \( 1 + (0.976 - 0.217i)T \)
71 \( 1 + (0.920 + 0.391i)T \)
73 \( 1 + (0.457 + 0.889i)T \)
79 \( 1 + (0.768 + 0.639i)T \)
83 \( 1 + (0.109 - 0.994i)T \)
89 \( 1 + (0.791 - 0.611i)T \)
97 \( 1 + (0.994 - 0.109i)T \)
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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

−27.459499082774292041745808017065, −26.43308587682349588216620362496, −25.29161515743166765565235845937, −24.69492982798585031840479534197, −23.174977978121588263915019819814, −22.15314457182705952533849195716, −21.07768518723373720494659847554, −20.23392173466842040257395466250, −19.607481429685536166747331050327, −18.17536490838195389644343299112, −17.34506525803727330008488087363, −16.244428020041970891565044193387, −15.58729596976716232396681779537, −14.865381407632626665234123779846, −12.57303987465119853272394594815, −11.946993698336833432084020500626, −10.876149738619411113893714425942, −9.744111252285081276146379791210, −8.863668999918951133645223457682, −8.12866868751218236234121529230, −6.46801230437797842094046525268, −5.11139033657143724361576627998, −3.815708659965258876284518403154, −2.34544386525984887457939347210, −0.37840788966093628142479782895, 0.908660338898299403827862803843, 2.34686652217418272537700433876, 3.94013235291461607028905793930, 6.324590983794286682347707052068, 6.692559852505190037018738090585, 7.84478009005509531750025088535, 8.653242402964475506481305356104, 10.33069602802634798288178344458, 11.225183376834228827223953896130, 11.8784398571774794591236823100, 13.55263766415892701169804235807, 14.43422076316115255037028814586, 15.758327238898643493887325761151, 16.871223551602816360980517287400, 17.53362642605584816618997243077, 18.72985035633237327257364807104, 19.37365043418157721128165729031, 19.89534362343707669959483471796, 21.489912534007018897335963343675, 22.90148134035489562111640209091, 23.85522090327284807467497128896, 24.26153456413777304703013114611, 25.731348563743946293166875584404, 26.354935486373720578787550755129, 27.204053251558561349859171668820

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