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Properties

Label 1-547-547.169-r0-0-0
Degree $1$
Conductor $547$
Sign $0.207 + 0.978i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (0.955 + 0.294i)2-s + (−0.900 + 0.433i)3-s + (0.826 + 0.563i)4-s + (−0.988 − 0.149i)5-s + (−0.988 + 0.149i)6-s + (0.826 − 0.563i)7-s + (0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.900 − 0.433i)10-s + (−0.5 + 0.866i)11-s + (−0.988 − 0.149i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (0.955 − 0.294i)15-s + (0.365 + 0.930i)16-s + (0.365 + 0.930i)17-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)2-s + (−0.900 + 0.433i)3-s + (0.826 + 0.563i)4-s + (−0.988 − 0.149i)5-s + (−0.988 + 0.149i)6-s + (0.826 − 0.563i)7-s + (0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.900 − 0.433i)10-s + (−0.5 + 0.866i)11-s + (−0.988 − 0.149i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (0.955 − 0.294i)15-s + (0.365 + 0.930i)16-s + (0.365 + 0.930i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.207 + 0.978i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.207 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.298738156 + 1.052206301i\)
\(L(\frac12)\) \(\approx\) \(1.298738156 + 1.052206301i\)
\(L(1)\) \(\approx\) \(1.244213891 + 0.4907527752i\)
\(L(1)\) \(\approx\) \(1.244213891 + 0.4907527752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.955 + 0.294i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (0.826 - 0.563i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (0.365 + 0.930i)T \)
19 \( 1 + (0.955 - 0.294i)T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.826 - 0.563i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.733 - 0.680i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.365 + 0.930i)T \)
67 \( 1 + (0.365 + 0.930i)T \)
71 \( 1 + (0.365 + 0.930i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (-0.900 - 0.433i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (0.955 - 0.294i)T \)
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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

−23.12392513051150897306064637427, −22.529559742934538686517080639233, −21.624565178027529636440591895752, −21.00097885272668589164864719979, −19.8713851295732938185333926502, −18.79595210956841205168461473182, −18.62448752239182721454370204493, −17.19193317067463655647738422858, −16.12216410012168836759588556173, −15.71018628792475816919668437698, −14.52699862480766495981964259785, −13.84436702294832418261893273515, −12.63468829173168901832188086737, −11.986416990589356634393596444468, −11.30626297501003101672515840975, −10.90883355950678963109439526186, −9.451896098636321524966586616134, −7.79026917473700576571012733757, −7.35254052426006172073555151756, −6.13456246719344195270845520600, −5.16856765643024573314367024732, −4.65110736957606063829350227710, −3.30070162866150809177325100582, −2.20014984135407185133793969760, −0.82914650498373358720078306279, 1.277229727847766102657472685603, 3.05127164260459032016358575208, 4.07952865216024251051425399556, 4.92429445875221652209195243471, 5.310167476193110675961292671373, 6.90184626167307179065287568635, 7.4169844185232850695556055269, 8.37694248255828566219841860223, 10.11728590111916110591371226893, 10.86079568557630956518116818006, 11.64125272138210631359112977986, 12.42194051477168816889527898839, 13.04241503598273499052040402415, 14.59976145756620657453462630119, 15.02943470480148858819964004211, 15.82269476702296890038160345563, 16.71000097070725629598097040903, 17.36451485916932601588321915199, 18.25736360563491477769114442303, 19.824974346289804402650720379686, 20.3828192274432782255514081849, 21.188868912890915217630131603561, 22.1139783462891298604342868095, 22.84722245872094118853113927214, 23.57399621474333280134393032334

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