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Properties

Label 1-547-547.209-r0-0-0
Degree $1$
Conductor $547$
Sign $0.0322 - 0.999i$
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  + (−0.952 − 0.305i)2-s + (−0.222 − 0.974i)3-s + (0.813 + 0.582i)4-s + (−0.596 + 0.802i)5-s + (−0.0862 + 0.996i)6-s + (−0.832 − 0.553i)7-s + (−0.596 − 0.802i)8-s + (−0.900 + 0.433i)9-s + (0.813 − 0.582i)10-s + (−0.970 + 0.239i)11-s + (0.386 − 0.922i)12-s + (0.623 + 0.781i)13-s + (0.623 + 0.781i)14-s + (0.915 + 0.402i)15-s + (0.322 + 0.946i)16-s + (0.978 − 0.205i)17-s + ⋯
L(s)  = 1  + (−0.952 − 0.305i)2-s + (−0.222 − 0.974i)3-s + (0.813 + 0.582i)4-s + (−0.596 + 0.802i)5-s + (−0.0862 + 0.996i)6-s + (−0.832 − 0.553i)7-s + (−0.596 − 0.802i)8-s + (−0.900 + 0.433i)9-s + (0.813 − 0.582i)10-s + (−0.970 + 0.239i)11-s + (0.386 − 0.922i)12-s + (0.623 + 0.781i)13-s + (0.623 + 0.781i)14-s + (0.915 + 0.402i)15-s + (0.322 + 0.946i)16-s + (0.978 − 0.205i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0322 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0322 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.0322 - 0.999i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.0322 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3262965239 - 0.3159253488i\)
\(L(\frac12)\) \(\approx\) \(0.3262965239 - 0.3159253488i\)
\(L(1)\) \(\approx\) \(0.4698307295 - 0.1648203811i\)
\(L(1)\) \(\approx\) \(0.4698307295 - 0.1648203811i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.952 - 0.305i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.596 + 0.802i)T \)
7 \( 1 + (-0.832 - 0.553i)T \)
11 \( 1 + (-0.970 + 0.239i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (0.978 - 0.205i)T \)
19 \( 1 + (-0.700 + 0.713i)T \)
23 \( 1 + (0.997 - 0.0689i)T \)
29 \( 1 + (-0.154 - 0.987i)T \)
31 \( 1 + (-0.994 + 0.103i)T \)
37 \( 1 + (-0.868 - 0.495i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.479 - 0.877i)T \)
47 \( 1 + (0.568 + 0.822i)T \)
53 \( 1 + (0.915 + 0.402i)T \)
59 \( 1 + (0.120 - 0.992i)T \)
61 \( 1 + (0.725 + 0.688i)T \)
67 \( 1 + (0.386 - 0.922i)T \)
71 \( 1 + (-0.539 - 0.842i)T \)
73 \( 1 + (0.915 - 0.402i)T \)
79 \( 1 + (-0.479 - 0.877i)T \)
83 \( 1 + (0.885 - 0.464i)T \)
89 \( 1 + (0.725 - 0.688i)T \)
97 \( 1 + (-0.418 - 0.908i)T \)
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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

−23.41728190267335432130471201180, −23.10321735138453247684984893155, −21.67472138861837041853880084558, −20.922556879886241624524472535708, −20.21686115982284641509010165490, −19.38791830368491237221536984515, −18.57256780378442834586948393503, −17.54759642850114196957434474229, −16.535576754784983182972285756804, −16.18024696410577181126559394905, −15.38332381182848352676308499869, −14.879803128880477792102602983941, −13.14518585171235616447494700295, −12.307476734296477160259906011389, −11.17900621255931612908620224018, −10.532683979970984192556405100729, −9.579698246397930577536257593661, −8.75458692221564698268910987082, −8.2222285302800947971299157994, −6.9361559662972454042514091666, −5.566870623936801562048259516701, −5.28342279400105226365794422507, −3.62617539427701946379155792944, −2.73465325557319029904879298941, −0.81863203295584105526900922857, 0.47988274782588447220754988685, 1.926194077027174813267890662391, 2.94573703385662410583453446639, 3.83936702786331423450882523599, 5.84107497592626027758770199924, 6.75781685195498275767915806276, 7.39075625522601365025640785049, 8.05664542688097965808373633079, 9.22212298078145446300594296775, 10.46291857694563960685519420448, 10.88811397959225172355083069932, 11.95455736250747916346263689226, 12.65868516441163716455511185668, 13.58353084552627992448803187168, 14.722041773252118972846972850321, 15.89402333976683767958657868372, 16.56973364925272397573847413930, 17.431430387160334754505909357830, 18.515129728038054584947999180820, 18.88179154674952256432747444795, 19.38709628783798918891855034745, 20.45034973659039196073398150239, 21.26188671551267856703830211680, 22.613047650088807424379720969518, 23.252203081966199303901004393549

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