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Properties

Label 1-547-547.115-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.999 + 0.0159i$
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  + (0.658 − 0.752i)2-s + (0.623 − 0.781i)3-s + (−0.131 − 0.991i)4-s + (−0.0632 − 0.997i)5-s + (−0.177 − 0.984i)6-s + (0.968 + 0.250i)7-s + (−0.832 − 0.553i)8-s + (−0.222 − 0.974i)9-s + (−0.792 − 0.609i)10-s + (−0.996 − 0.0804i)11-s + (−0.857 − 0.514i)12-s + (0.0747 + 0.997i)13-s + (0.826 − 0.563i)14-s + (−0.819 − 0.572i)15-s + (−0.965 + 0.261i)16-s + (−0.976 − 0.216i)17-s + ⋯
L(s)  = 1  + (0.658 − 0.752i)2-s + (0.623 − 0.781i)3-s + (−0.131 − 0.991i)4-s + (−0.0632 − 0.997i)5-s + (−0.177 − 0.984i)6-s + (0.968 + 0.250i)7-s + (−0.832 − 0.553i)8-s + (−0.222 − 0.974i)9-s + (−0.792 − 0.609i)10-s + (−0.996 − 0.0804i)11-s + (−0.857 − 0.514i)12-s + (0.0747 + 0.997i)13-s + (0.826 − 0.563i)14-s + (−0.819 − 0.572i)15-s + (−0.965 + 0.261i)16-s + (−0.976 − 0.216i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.999 + 0.0159i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.999 + 0.0159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01758814583 - 2.209136113i\)
\(L(\frac12)\) \(\approx\) \(0.01758814583 - 2.209136113i\)
\(L(1)\) \(\approx\) \(0.9681821816 - 1.386938823i\)
\(L(1)\) \(\approx\) \(0.9681821816 - 1.386938823i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.658 - 0.752i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (-0.0632 - 0.997i)T \)
7 \( 1 + (0.968 + 0.250i)T \)
11 \( 1 + (-0.996 - 0.0804i)T \)
13 \( 1 + (0.0747 + 0.997i)T \)
17 \( 1 + (-0.976 - 0.216i)T \)
19 \( 1 + (0.469 - 0.882i)T \)
23 \( 1 + (-0.244 + 0.969i)T \)
29 \( 1 + (0.813 - 0.582i)T \)
31 \( 1 + (0.915 - 0.402i)T \)
37 \( 1 + (0.999 - 0.0230i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.577 + 0.816i)T \)
47 \( 1 + (0.948 - 0.316i)T \)
53 \( 1 + (0.905 - 0.423i)T \)
59 \( 1 + (-0.0402 - 0.999i)T \)
61 \( 1 + (0.586 + 0.809i)T \)
67 \( 1 + (0.874 - 0.484i)T \)
71 \( 1 + (-0.332 + 0.942i)T \)
73 \( 1 + (-0.819 + 0.572i)T \)
79 \( 1 + (-0.418 - 0.908i)T \)
83 \( 1 + (-0.632 + 0.774i)T \)
89 \( 1 + (-0.994 - 0.103i)T \)
97 \( 1 + (0.932 + 0.359i)T \)
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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

−23.67766537497541707523315038575, −22.928796194614034782870570419046, −22.13730033554993139015082148135, −21.46166667978713236032638149595, −20.6295954225843972714554857405, −19.99690840017462151019933176551, −18.43191074554853768376543933713, −17.93176537632228393549815779220, −16.87058409294145869961593889223, −15.75237431119741076167569374871, −15.29930913680862349340477147907, −14.52685724869481689938544312511, −13.921088839247468997162225163486, −13.06174999790357650874665850564, −11.75212710045302127425552892717, −10.68299664938797039901539377271, −10.18331013216350047604691762409, −8.53442962379683708555125617143, −8.037286009323045762551951843041, −7.21577968001067463544512710643, −5.94706549611965795177467790072, −4.964515413687345247665859268665, −4.167756020655754927486903358451, −3.077161845523888535640137948712, −2.370106248364701919372245422000, 0.85741968379379398031956933103, 1.939286820682640055330882816831, 2.62505264359679921898479205598, 4.1121612028013532566868264257, 4.8748688279011755167284499531, 5.844158346896334203553979092100, 7.1057467372655195493435221460, 8.261663771430121738896690227579, 8.94822436423555982121332622547, 9.86551924971947536343307102905, 11.49582340887924290216285971589, 11.65833427569189319258628873421, 12.829971654526350722161370569371, 13.53711683225084975326826065082, 13.98445841926290334836077580054, 15.24472655391611152898964567265, 15.79139703505582349534138843287, 17.42634229338914089762509241782, 18.097377003774712567274042566353, 18.98245328400261904467413909917, 19.82972010627276992990422343374, 20.471595874662122578848156679440, 21.18211337787221635018399042575, 21.767276012073547503347741881584, 23.328989343922975180773336999818

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