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Properties

Label 1-547-547.166-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.477 - 0.878i$
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  + (0.838 + 0.544i)2-s + (−0.222 + 0.974i)3-s + (0.407 + 0.913i)4-s + (−0.778 + 0.627i)5-s + (−0.717 + 0.696i)6-s + (−0.910 + 0.413i)7-s + (−0.154 + 0.987i)8-s + (−0.900 − 0.433i)9-s + (−0.994 + 0.103i)10-s + (−0.845 + 0.534i)11-s + (−0.980 + 0.194i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (−0.439 − 0.898i)15-s + (−0.667 + 0.744i)16-s + (0.469 − 0.882i)17-s + ⋯
L(s)  = 1  + (0.838 + 0.544i)2-s + (−0.222 + 0.974i)3-s + (0.407 + 0.913i)4-s + (−0.778 + 0.627i)5-s + (−0.717 + 0.696i)6-s + (−0.910 + 0.413i)7-s + (−0.154 + 0.987i)8-s + (−0.900 − 0.433i)9-s + (−0.994 + 0.103i)10-s + (−0.845 + 0.534i)11-s + (−0.980 + 0.194i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (−0.439 − 0.898i)15-s + (−0.667 + 0.744i)16-s + (0.469 − 0.882i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.477 - 0.878i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.477 - 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5299316962 + 0.8909344724i\)
\(L(\frac12)\) \(\approx\) \(-0.5299316962 + 0.8909344724i\)
\(L(1)\) \(\approx\) \(0.5616087199 + 0.9126953712i\)
\(L(1)\) \(\approx\) \(0.5616087199 + 0.9126953712i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.838 + 0.544i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (-0.778 + 0.627i)T \)
7 \( 1 + (-0.910 + 0.413i)T \)
11 \( 1 + (-0.845 + 0.534i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (0.469 - 0.882i)T \)
19 \( 1 + (0.924 + 0.381i)T \)
23 \( 1 + (0.490 - 0.871i)T \)
29 \( 1 + (0.725 + 0.688i)T \)
31 \( 1 + (-0.0172 - 0.999i)T \)
37 \( 1 + (-0.819 - 0.572i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.177 + 0.984i)T \)
47 \( 1 + (-0.632 + 0.774i)T \)
53 \( 1 + (-0.558 + 0.829i)T \)
59 \( 1 + (0.278 - 0.960i)T \)
61 \( 1 + (-0.991 - 0.126i)T \)
67 \( 1 + (0.658 + 0.752i)T \)
71 \( 1 + (-0.937 + 0.349i)T \)
73 \( 1 + (-0.439 + 0.898i)T \)
79 \( 1 + (0.940 - 0.338i)T \)
83 \( 1 + (-0.996 + 0.0804i)T \)
89 \( 1 + (0.386 - 0.922i)T \)
97 \( 1 + (-0.944 + 0.327i)T \)
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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

−23.0389455039157426218656890644, −22.23701480450306292725196774420, −21.04047204903286684816151984254, −20.216977721854981728915043169173, −19.45097260208616077485698923758, −19.0814333503745484054346583885, −17.9665699969552382589859208790, −16.796331253206467793853805780873, −15.86616739339257912470099069444, −15.26607717954500967176097198741, −13.70448871331333251289890255235, −13.378855760065642019864360981720, −12.53557879247342936516859581586, −11.95974280178030003530406646745, −10.92787932300587328178209606619, −10.1602321936276079082462883029, −8.701069057393651482695990393481, −7.70876199098185460443838392023, −6.82153704613246334941910569090, −5.7026142809163672373680185400, −5.08550580478552314224804601467, −3.50276444836195600288815896669, −3.05876607317035167044159655125, −1.41989853699331912332560633801, −0.42834006778028178066434081005, 2.74378397633939887002493455362, 3.24568356533945380715999048612, 4.31384720424522619929568404077, 5.11816947367022286742313770044, 6.20431229616143694142289888021, 7.00453191948260586733747037098, 8.041142431300390952003846021096, 9.1774356857246316838579487439, 10.1687948199369397238666575815, 11.25046861214245632809575952262, 11.91660949167285756394037874523, 12.77882591166626283843875769918, 14.051246834518493088898709991673, 14.69487894446713974906607217469, 15.67490958411267751442812697460, 16.02485325342084736915941623152, 16.64956222148713421040198461410, 18.02937662412545867697624389588, 18.83973980376133398717857638201, 20.113832498866945176886865847450, 20.78504133376153235503913605770, 21.68307716836204750497503887872, 22.46685912751475217325595307540, 23.01669454505463168313785847223, 23.48930884308891641146388467799

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