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Properties

Label 1-547-547.9-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.419 + 0.907i$
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  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)3-s + (0.623 + 0.781i)4-s + (−0.222 + 0.974i)5-s + (−0.222 − 0.974i)6-s + (0.623 − 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.222 + 0.974i)9-s + (0.623 − 0.781i)10-s + 11-s + (−0.222 + 0.974i)12-s + (−0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + (−0.900 + 0.433i)15-s + (−0.222 + 0.974i)16-s + (−0.222 + 0.974i)17-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)3-s + (0.623 + 0.781i)4-s + (−0.222 + 0.974i)5-s + (−0.222 − 0.974i)6-s + (0.623 − 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.222 + 0.974i)9-s + (0.623 − 0.781i)10-s + 11-s + (−0.222 + 0.974i)12-s + (−0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + (−0.900 + 0.433i)15-s + (−0.222 + 0.974i)16-s + (−0.222 + 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.419 + 0.907i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.419 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5075835761 + 0.7937820496i\)
\(L(\frac12)\) \(\approx\) \(0.5075835761 + 0.7937820496i\)
\(L(1)\) \(\approx\) \(0.7734230023 + 0.3196324840i\)
\(L(1)\) \(\approx\) \(0.7734230023 + 0.3196324840i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.900 - 0.433i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 + (-0.900 + 0.433i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + (0.623 + 0.781i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 + (0.623 - 0.781i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−23.64199871256878953361897044376, −22.36415818153868054829400452165, −21.03858328388069900364894744833, −20.304604120389578190205416055644, −19.61722311286627697265030160977, −19.02171724352677736573185716412, −17.89013874086790719385695841216, −17.49734960091622404992691458036, −16.47852769920225048716641730477, −15.48551361544287926169654098965, −14.69693005058000464658663031742, −14.010937729144625457752434266095, −12.59212649102645180240001400083, −12.040637568956221063157473856551, −11.128894592359766608005755381513, −9.464609178845929642416051617247, −9.10222599866606919294844505066, −8.2250987619048512932016558966, −7.57961254136804783604975138654, −6.51980336042046073118087446585, −5.51797769807964922440913175739, −4.39448546927920008907148556816, −2.578865516558551695170558160192, −1.79217380166133979205860780147, −0.60004199049294047929869250368, 1.69095720218256957479565991958, 2.579247999982368380630310722946, 3.91008169796780730240866010648, 4.18817976025670433826072276271, 6.23241125706207664613949787043, 7.28188555997472539214295408345, 8.00429790845240248960695314965, 8.90067421310164040916096132039, 10.01865379686767944730964940434, 10.42342810759391246560519302477, 11.3140747275097749124491153713, 12.10835996076804769204671284338, 13.637571263402446042718696498070, 14.50349129289409812178447910527, 15.09266137165243018817170708293, 16.14239527904157858554138284611, 17.22779019904601494397423656544, 17.47835390850366408632696968273, 19.10893216311666882033090056186, 19.305857968234535902040281465615, 20.16050880006824094411079446017, 21.04541511723171784942583819815, 21.83218167726794807777695228178, 22.37560356814675193086825883748, 23.71531095308128850381383464760

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