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Properties

Label 1-547-547.119-r0-0-0
Degree $1$
Conductor $547$
Sign $0.534 + 0.845i$
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  + (−0.976 + 0.216i)2-s + (−0.900 − 0.433i)3-s + (0.905 − 0.423i)4-s + (−0.131 + 0.991i)5-s + (0.973 + 0.228i)6-s + (0.863 − 0.504i)7-s + (−0.792 + 0.609i)8-s + (0.623 + 0.781i)9-s + (−0.0862 − 0.996i)10-s + (−0.919 − 0.391i)11-s + (−0.999 − 0.0115i)12-s + (0.955 + 0.294i)13-s + (−0.733 + 0.680i)14-s + (0.548 − 0.835i)15-s + (0.641 − 0.767i)16-s + (0.0287 + 0.999i)17-s + ⋯
L(s)  = 1  + (−0.976 + 0.216i)2-s + (−0.900 − 0.433i)3-s + (0.905 − 0.423i)4-s + (−0.131 + 0.991i)5-s + (0.973 + 0.228i)6-s + (0.863 − 0.504i)7-s + (−0.792 + 0.609i)8-s + (0.623 + 0.781i)9-s + (−0.0862 − 0.996i)10-s + (−0.919 − 0.391i)11-s + (−0.999 − 0.0115i)12-s + (0.955 + 0.294i)13-s + (−0.733 + 0.680i)14-s + (0.548 − 0.835i)15-s + (0.641 − 0.767i)16-s + (0.0287 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.534 + 0.845i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.534 + 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5709361684 + 0.3145520072i\)
\(L(\frac12)\) \(\approx\) \(0.5709361684 + 0.3145520072i\)
\(L(1)\) \(\approx\) \(0.5843362288 + 0.1054940408i\)
\(L(1)\) \(\approx\) \(0.5843362288 + 0.1054940408i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.976 + 0.216i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.131 + 0.991i)T \)
7 \( 1 + (0.863 - 0.504i)T \)
11 \( 1 + (-0.919 - 0.391i)T \)
13 \( 1 + (0.955 + 0.294i)T \)
17 \( 1 + (0.0287 + 0.999i)T \)
19 \( 1 + (0.999 - 0.0230i)T \)
23 \( 1 + (-0.332 + 0.942i)T \)
29 \( 1 + (0.915 - 0.402i)T \)
31 \( 1 + (0.962 - 0.272i)T \)
37 \( 1 + (-0.944 - 0.327i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.958 + 0.283i)T \)
47 \( 1 + (-0.0402 - 0.999i)T \)
53 \( 1 + (-0.998 - 0.0575i)T \)
59 \( 1 + (-0.200 - 0.979i)T \)
61 \( 1 + (-0.439 + 0.898i)T \)
67 \( 1 + (0.490 + 0.871i)T \)
71 \( 1 + (0.838 + 0.544i)T \)
73 \( 1 + (0.548 + 0.835i)T \)
79 \( 1 + (0.725 + 0.688i)T \)
83 \( 1 + (0.278 - 0.960i)T \)
89 \( 1 + (0.997 + 0.0689i)T \)
97 \( 1 + (0.586 + 0.809i)T \)
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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

−23.37249798700030563522059453834, −22.309836360399261620724537306846, −21.12775679897024215587332720166, −20.79511075841950406885743882775, −20.197528445948040781818662336128, −18.67438070549624265732018801557, −18.08014504026461998805969785852, −17.5294806170250423880301873447, −16.52352979586295798000852387236, −15.7583084181217224776429503238, −15.46892121958841903957434514168, −13.75214731944932602268092772783, −12.358943638383855189057847944, −12.04858171200633727691254446987, −11.07346297825613890657459967223, −10.30130969552484060246356871953, −9.34744678534856016182699736408, −8.47243831049768278371820455810, −7.71678813661660948116833147992, −6.448818584415688542414323841237, −5.30219789974494816216262017292, −4.70211644320926218664224718141, −3.182196360657523213872036839261, −1.6984829460274084569778181379, −0.65888566327619274384804287077, 1.085806896620298234958073416244, 2.075338135908949861232538497396, 3.50627784654027667445477839226, 5.113379873144603833760164576823, 6.08327541862886246210528254321, 6.81868853350955885890567822519, 7.81999436016976662034691679669, 8.24055868147892386096927089363, 10.01725960168282885159859996689, 10.54257358785591698247851019898, 11.35030566631579681282353964293, 11.78366982707194571147375923468, 13.39282380931418260835190225136, 14.15809244800294244262136238030, 15.43739094950054267489575842211, 15.944781488704054616540025836114, 17.0699611201240574272976446508, 17.75470484573157435332500995415, 18.33560019460308217424268785122, 18.95395224862136819458310250675, 19.88188905611587524679485401406, 21.11212979398273217559017808588, 21.67633803583242879613329689470, 23.11601142782632528579809492056, 23.52955178747325394513187225372

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