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Properties

Label 1-547-547.173-r1-0-0
Degree $1$
Conductor $547$
Sign $-0.626 + 0.779i$
Analytic cond. $58.7833$
Root an. cond. $58.7833$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.626 + 0.779i$
Analytic conductor: \(58.7833\)
Root analytic conductor: \(58.7833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (1:\ ),\ -0.626 + 0.779i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08946151788 + 0.1865805189i\)
\(L(\frac12)\) \(\approx\) \(0.08946151788 + 0.1865805189i\)
\(L(1)\) \(\approx\) \(1.402226344 - 0.4191211558i\)
\(L(1)\) \(\approx\) \(1.402226344 - 0.4191211558i\)

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

−23.14424062657738389634932079595, −21.89481597442334203219450494148, −21.22444583543475709170182228291, −20.45638888861535671229472084605, −19.90027666534893056651677636281, −18.96118755193485399054463594959, −17.3771564656154150298098976426, −16.69938888718827678363076397586, −15.81882110960609651164423454169, −15.41403471896559478374786593125, −14.34567293986849912326963740744, −13.268787785330318111192073379296, −13.08644233694797767961161346735, −12.11285599513942065949083951280, −10.57553900164475277267961230276, −9.866864339397213871479487610050, −8.44534085065751467594264122008, −8.07731669572649701062506703440, −7.11502825684916289895097198664, −5.691055034895346298065279864650, −4.94927597425385444096033145300, −3.80901504978949339672800174598, −3.322474577739729264216897204355, −1.955477534942592807648350019057, −0.02855690857982993696818667556, 1.91793920617590936010007435129, 2.60410957488319568249634890321, 3.31685269695498509350090462483, 4.35704827681090584081630874990, 5.76660769276098437402863769529, 6.70605675408625243604018508344, 7.38102355693611613482181457892, 8.74578902260381710261449969254, 9.80458103714385324029548275740, 10.44403565159978666869556290366, 11.72225831918536070997282103859, 12.41222037274190964600440167197, 13.2405484404701822300147543192, 14.0978357717267089406548068081, 14.80641043226638783286694591770, 15.43193378091431158560205710583, 16.33672314324978627920314563499, 18.248370247218057877562855656128, 18.73928091985568136783599012077, 19.18853092405287595419060911881, 20.16463759384087036032323278788, 21.03142375922230599543264153478, 21.76908818317523495854300758832, 22.54536354588558848333707012455, 23.509441873161186788586179805277

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