Properties

Label 1-547-547.490-r0-0-0
Degree $1$
Conductor $547$
Sign $0.640 + 0.768i$
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  + (−0.267 − 0.963i)2-s + (−0.222 + 0.974i)3-s + (−0.857 + 0.514i)4-s + (−0.958 − 0.283i)5-s + (0.998 − 0.0460i)6-s + (0.407 − 0.913i)7-s + (0.725 + 0.688i)8-s + (−0.900 − 0.433i)9-s + (−0.0172 + 0.999i)10-s + (−0.996 + 0.0804i)11-s + (−0.311 − 0.950i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (0.490 − 0.871i)15-s + (0.469 − 0.882i)16-s + (0.00575 − 0.999i)17-s + ⋯
L(s)  = 1  + (−0.267 − 0.963i)2-s + (−0.222 + 0.974i)3-s + (−0.857 + 0.514i)4-s + (−0.958 − 0.283i)5-s + (0.998 − 0.0460i)6-s + (0.407 − 0.913i)7-s + (0.725 + 0.688i)8-s + (−0.900 − 0.433i)9-s + (−0.0172 + 0.999i)10-s + (−0.996 + 0.0804i)11-s + (−0.311 − 0.950i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (0.490 − 0.871i)15-s + (0.469 − 0.882i)16-s + (0.00575 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.640 + 0.768i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (490, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.640 + 0.768i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4532192829 + 0.2121925704i\)
\(L(\frac12)\) \(\approx\) \(0.4532192829 + 0.2121925704i\)
\(L(1)\) \(\approx\) \(0.5952111275 - 0.07757516741i\)
\(L(1)\) \(\approx\) \(0.5952111275 - 0.07757516741i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.267 - 0.963i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (-0.958 - 0.283i)T \)
7 \( 1 + (0.407 - 0.913i)T \)
11 \( 1 + (-0.996 + 0.0804i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (0.00575 - 0.999i)T \)
19 \( 1 + (-0.806 - 0.591i)T \)
23 \( 1 + (0.641 + 0.767i)T \)
29 \( 1 + (0.386 - 0.922i)T \)
31 \( 1 + (0.256 + 0.966i)T \)
37 \( 1 + (-0.244 + 0.969i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.998 - 0.0575i)T \)
47 \( 1 + (0.948 + 0.316i)T \)
53 \( 1 + (-0.999 + 0.0115i)T \)
59 \( 1 + (-0.0402 + 0.999i)T \)
61 \( 1 + (0.658 + 0.752i)T \)
67 \( 1 + (-0.667 + 0.744i)T \)
71 \( 1 + (0.993 - 0.114i)T \)
73 \( 1 + (0.490 + 0.871i)T \)
79 \( 1 + (0.449 + 0.893i)T \)
83 \( 1 + (-0.632 - 0.774i)T \)
89 \( 1 + (0.322 + 0.946i)T \)
97 \( 1 + (-0.684 + 0.729i)T \)
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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

−23.54206538290617590290620593876, −22.80749129481351933229315349367, −21.979423231051004155677886367696, −20.62253894040464793426379129714, −19.45745254089344385743008388339, −18.737488584329421838134510005636, −18.3598433177041810291402448679, −17.468325815048143778064887612548, −16.5410999933635020240961060706, −15.472319180183756623787089480294, −15.02602269236207466491921484908, −14.09010011280343079062066600814, −12.73755535231949520337650243594, −12.55938651884144226335106158776, −11.06273208607220946152234176928, −10.46329830190385300206939852639, −8.633143377485253397496406221658, −8.26904173211328946563649928099, −7.58027525460249834524337830892, −6.50529142481100998200576342896, −5.70804428926284261226451895690, −4.83286270377631294754243171811, −3.33822045016201434798559524539, −1.97278921489216761783015105119, −0.37359174454378545319136923401, 1.01724059426156170305475546676, 2.73850202893578235904985400875, 3.70166747062585016715104936457, 4.59388997196374834357683283239, 5.00038040013097434029996419713, 6.959594315921505442856832269000, 8.06519768014988963719952581279, 8.80270248677850479500402920187, 9.81257332614047100823984522847, 10.67361143863203312730225901617, 11.34784284000746207768648911540, 11.89432136297893277437373939825, 13.2124222642141330430805991399, 13.95332466475169227581871703586, 15.13848824255888349608999294500, 16.02104094565908573959968967026, 16.809496155371339152892108995014, 17.54880857001202729143050080691, 18.62861219652221517613047923571, 19.54591777037063406164683463633, 20.28957662652032285297655663368, 20.96974792368764770148728847799, 21.45282911108557068622001023971, 22.66399674389889361793712111477, 23.43163220964205547279533826196

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