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Properties

Label 1-547-547.199-r0-0-0
Degree $1$
Conductor $547$
Sign $0.189 - 0.981i$
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  + (0.692 − 0.721i)2-s + 3-s + (−0.0402 − 0.999i)4-s + (0.948 + 0.316i)5-s + (0.692 − 0.721i)6-s + (0.278 − 0.960i)7-s + (−0.748 − 0.663i)8-s + 9-s + (0.885 − 0.464i)10-s + (0.987 + 0.160i)11-s + (−0.0402 − 0.999i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.948 + 0.316i)15-s + (−0.996 + 0.0804i)16-s + (−0.632 − 0.774i)17-s + ⋯
L(s)  = 1  + (0.692 − 0.721i)2-s + 3-s + (−0.0402 − 0.999i)4-s + (0.948 + 0.316i)5-s + (0.692 − 0.721i)6-s + (0.278 − 0.960i)7-s + (−0.748 − 0.663i)8-s + 9-s + (0.885 − 0.464i)10-s + (0.987 + 0.160i)11-s + (−0.0402 − 0.999i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.948 + 0.316i)15-s + (−0.996 + 0.0804i)16-s + (−0.632 − 0.774i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.189 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.537303685 - 2.093898269i\)
\(L(\frac12)\) \(\approx\) \(2.537303685 - 2.093898269i\)
\(L(1)\) \(\approx\) \(2.045121794 - 1.078435994i\)
\(L(1)\) \(\approx\) \(2.045121794 - 1.078435994i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.692 - 0.721i)T \)
3 \( 1 + T \)
5 \( 1 + (0.948 + 0.316i)T \)
7 \( 1 + (0.278 - 0.960i)T \)
11 \( 1 + (0.987 + 0.160i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.632 - 0.774i)T \)
19 \( 1 + (-0.996 - 0.0804i)T \)
23 \( 1 + (-0.919 + 0.391i)T \)
29 \( 1 + (0.120 + 0.992i)T \)
31 \( 1 + (0.568 + 0.822i)T \)
37 \( 1 + (-0.919 - 0.391i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.845 + 0.534i)T \)
47 \( 1 + (0.799 - 0.600i)T \)
53 \( 1 + (-0.200 - 0.979i)T \)
59 \( 1 + (-0.996 + 0.0804i)T \)
61 \( 1 + (0.692 - 0.721i)T \)
67 \( 1 + (-0.845 + 0.534i)T \)
71 \( 1 + (0.428 + 0.903i)T \)
73 \( 1 + (0.948 - 0.316i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (-0.200 - 0.979i)T \)
89 \( 1 + (-0.970 + 0.239i)T \)
97 \( 1 + (0.987 - 0.160i)T \)
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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

−24.06453787918663545276886560129, −22.35746189774575841920125396446, −22.015718462958197045947557192995, −21.17827742388350641453521649576, −20.51646683320850697486278779913, −19.484395939347885594963479854211, −18.41909826516537256575345222477, −17.44310602666480431346073959878, −16.88728848415777023106847231662, −15.44832722018007139909248139238, −15.14373621306818902984398908644, −14.19632916459956057984634027859, −13.56494843008259312509170771098, −12.629030333076692310642269624265, −12.05649504659820452349206301305, −10.4192884831696685047071139559, −9.282115484215591785638150495884, −8.604505727264136015438906246343, −7.96237203664893247977817245104, −6.542244746731281685632479980968, −5.94597530035741652780221448906, −4.783425817218510119912727929383, −3.88067486872010866216844015919, −2.579009498080623849800234884973, −1.93129891656746755504466521935, 1.488351987466834115363018312502, 2.06140524262969619284865657285, 3.231686591313207009220520410, 4.18982474587310556089286471623, 4.942812156115054778680833409348, 6.63635863821621828082774067885, 6.92692509557026216945062523088, 8.64549927379893375407926297985, 9.52660976338372188080122592438, 10.130472395357233465775852741067, 11.062951119597711948074817171544, 12.16433791898957573956569477449, 13.20125832649447669864206803385, 14.00021030671464901058049321216, 14.22714545821028161768738011718, 15.075132207225103823110600522421, 16.35184253162865590454940356627, 17.49508319111266453524103381903, 18.3548013595042853548823904757, 19.41578732455390363731538351857, 19.92505768747284705346819177388, 20.697961580339860149168198282127, 21.57138929608896167195479817430, 21.98273777936439168416585359165, 23.119892493861518628662967472781

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