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Properties

Label 1-547-547.215-r0-0-0
Degree $1$
Conductor $547$
Sign $0.346 - 0.938i$
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  + (−0.832 − 0.553i)2-s + (−0.900 − 0.433i)3-s + (0.386 + 0.922i)4-s + (0.188 + 0.982i)5-s + (0.509 + 0.860i)6-s + (0.725 + 0.688i)7-s + (0.188 − 0.982i)8-s + (0.623 + 0.781i)9-s + (0.386 − 0.922i)10-s + (−0.970 − 0.239i)11-s + (0.0517 − 0.998i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.256 − 0.966i)15-s + (−0.700 + 0.713i)16-s + (−0.792 − 0.609i)17-s + ⋯
L(s)  = 1  + (−0.832 − 0.553i)2-s + (−0.900 − 0.433i)3-s + (0.386 + 0.922i)4-s + (0.188 + 0.982i)5-s + (0.509 + 0.860i)6-s + (0.725 + 0.688i)7-s + (0.188 − 0.982i)8-s + (0.623 + 0.781i)9-s + (0.386 − 0.922i)10-s + (−0.970 − 0.239i)11-s + (0.0517 − 0.998i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.256 − 0.966i)15-s + (−0.700 + 0.713i)16-s + (−0.792 − 0.609i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.346 - 0.938i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.346 - 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4664229915 - 0.3250970208i\)
\(L(\frac12)\) \(\approx\) \(0.4664229915 - 0.3250970208i\)
\(L(1)\) \(\approx\) \(0.5460764368 - 0.1358446338i\)
\(L(1)\) \(\approx\) \(0.5460764368 - 0.1358446338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.832 - 0.553i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (0.188 + 0.982i)T \)
7 \( 1 + (0.725 + 0.688i)T \)
11 \( 1 + (-0.970 - 0.239i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.792 - 0.609i)T \)
19 \( 1 + (-0.994 + 0.103i)T \)
23 \( 1 + (0.675 - 0.736i)T \)
29 \( 1 + (-0.289 - 0.957i)T \)
31 \( 1 + (0.322 - 0.946i)T \)
37 \( 1 + (0.997 - 0.0689i)T \)
41 \( 1 + T \)
43 \( 1 + (0.962 + 0.272i)T \)
47 \( 1 + (0.568 - 0.822i)T \)
53 \( 1 + (0.256 - 0.966i)T \)
59 \( 1 + (0.120 + 0.992i)T \)
61 \( 1 + (-0.952 + 0.305i)T \)
67 \( 1 + (0.0517 - 0.998i)T \)
71 \( 1 + (0.851 - 0.524i)T \)
73 \( 1 + (0.256 + 0.966i)T \)
79 \( 1 + (0.962 + 0.272i)T \)
83 \( 1 + (0.885 + 0.464i)T \)
89 \( 1 + (-0.952 - 0.305i)T \)
97 \( 1 + (0.449 + 0.893i)T \)
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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

−23.83768022530745036901858274572, −23.18005969896920645337297733725, −21.610707043377661921846949015692, −21.05253264698864909319297007900, −20.22426528911736017479479707604, −19.26893471957911575799009033001, −18.11205064247251302782414720641, −17.40059647918316501448180065623, −16.98148698386758239789277660450, −16.144912718804921915101959427878, −15.43570964792830058131122482028, −14.44515844342857741677733095859, −13.26447199749576175661067958862, −12.27685936154992410656240172545, −11.005292578194520801721183295782, −10.720261163048366777423880938857, −9.534455223899338772305739743973, −8.83631135918600240807894887549, −7.756599200391172985859378209810, −6.83471784188073956429579809261, −5.7966777098824525778662574294, −4.84371945134255803491963021174, −4.338380704002490320609238013817, −1.99316023561216366668133062726, −0.9913375097066261265963068393, 0.54532326750650820804537540796, 2.31258079730375676579551776424, 2.54401140230856562099709870439, 4.35638575114169304001292970328, 5.61416279777456337548602707062, 6.500979369253479686346210967989, 7.59454112584315597534861269090, 8.139990435476620868199805835394, 9.49069564662998015933696504002, 10.66148483376195134710108727351, 10.89862564855859551438858667226, 11.7639582246157706460309557722, 12.75559904908161747719043161842, 13.46138032800585095418566548838, 15.02532386797952623871573837863, 15.64635876591619894575619153301, 16.85264200059069200763512570287, 17.65281299371950169409005063272, 18.25078379031602420872221221936, 18.676455362998485931957983402064, 19.565832413019639961085171390870, 20.921799181901173023487003202708, 21.41622890952634498474966109381, 22.41807715000377554284779141106, 22.88442613090869843422700047501

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