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Properties

Label 1-547-547.76-r0-0-0
Degree $1$
Conductor $547$
Sign $0.985 - 0.167i$
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  + (−0.890 + 0.454i)2-s + (−0.222 + 0.974i)3-s + (0.586 − 0.809i)4-s + (0.932 + 0.359i)5-s + (−0.244 − 0.969i)6-s + (0.0976 − 0.995i)7-s + (−0.154 + 0.987i)8-s + (−0.900 − 0.433i)9-s + (−0.994 + 0.103i)10-s + (−0.0402 − 0.999i)11-s + (0.658 + 0.752i)12-s + (−0.988 − 0.149i)13-s + (0.365 + 0.930i)14-s + (−0.558 + 0.829i)15-s + (−0.311 − 0.950i)16-s + (0.529 + 0.848i)17-s + ⋯
L(s)  = 1  + (−0.890 + 0.454i)2-s + (−0.222 + 0.974i)3-s + (0.586 − 0.809i)4-s + (0.932 + 0.359i)5-s + (−0.244 − 0.969i)6-s + (0.0976 − 0.995i)7-s + (−0.154 + 0.987i)8-s + (−0.900 − 0.433i)9-s + (−0.994 + 0.103i)10-s + (−0.0402 − 0.999i)11-s + (0.658 + 0.752i)12-s + (−0.988 − 0.149i)13-s + (0.365 + 0.930i)14-s + (−0.558 + 0.829i)15-s + (−0.311 − 0.950i)16-s + (0.529 + 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.985 - 0.167i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.985 - 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7952607846 - 0.06727297201i\)
\(L(\frac12)\) \(\approx\) \(0.7952607846 - 0.06727297201i\)
\(L(1)\) \(\approx\) \(0.7085519630 + 0.1540552521i\)
\(L(1)\) \(\approx\) \(0.7085519630 + 0.1540552521i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.890 + 0.454i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.932 + 0.359i)T \)
7 \( 1 + (0.0976 - 0.995i)T \)
11 \( 1 + (-0.0402 - 0.999i)T \)
13 \( 1 + (-0.988 - 0.149i)T \)
17 \( 1 + (0.529 + 0.848i)T \)
19 \( 1 + (-0.131 - 0.991i)T \)
23 \( 1 + (-0.999 + 0.0115i)T \)
29 \( 1 + (0.725 + 0.688i)T \)
31 \( 1 + (-0.0172 - 0.999i)T \)
37 \( 1 + (0.905 - 0.423i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.763 - 0.645i)T \)
47 \( 1 + (0.987 + 0.160i)T \)
53 \( 1 + (-0.439 - 0.898i)T \)
59 \( 1 + (0.692 + 0.721i)T \)
61 \( 1 + (0.605 - 0.795i)T \)
67 \( 1 + (-0.980 + 0.194i)T \)
71 \( 1 + (0.166 - 0.986i)T \)
73 \( 1 + (-0.558 - 0.829i)T \)
79 \( 1 + (0.940 - 0.338i)T \)
83 \( 1 + (0.428 - 0.903i)T \)
89 \( 1 + (0.386 - 0.922i)T \)
97 \( 1 + (0.756 + 0.654i)T \)
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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

−23.50815929017519464880902621096, −22.32577161118346032049983719079, −21.685029729157742388323372406563, −20.69663207979031507990494668103, −19.97423258107967017276886393134, −19.028486154515767735837417148905, −18.19211660529103753221950585774, −17.84356040230517954905037258814, −16.96361401744871630215347203788, −16.20728327209022911782768074947, −14.852954125242345622233549768753, −13.88077139566065678770144338039, −12.67905582862115985012242561108, −12.24080547787799686739936236193, −11.62407464275624255951530069531, −10.09279158518995136272965122383, −9.6507467511664968655285075940, −8.53868501052158658774591410613, −7.76778455032757770100749109209, −6.75156715059451950924367358597, −5.86144962036720168955479872328, −4.78296214648464411778405704466, −2.76049777942216612519257161095, −2.160981933206195448545108725623, −1.29093616844429552896838375101, 0.5984012852344834796304731649, 2.19156968026769410872139694475, 3.40065826295764636375503481607, 4.799960789487089196837664600218, 5.74254813959298432463602391739, 6.49509075785249700490776957388, 7.61272417697209810660338746022, 8.68382284265615719663886189193, 9.587029126742063497786931393666, 10.36988799014082871101038179458, 10.71243881846618670848930270645, 11.78237973245354354304947515523, 13.445346082771619816036962639319, 14.33046966845576855768047480997, 14.879031333660728696893491605977, 16.00730059071841661939476039792, 16.81203882466075186651304704674, 17.26492508398336613293161944495, 17.98299694807602283520647440543, 19.1943544171457127042007803389, 19.96563749524808066149130503236, 20.80075416397644584277585190438, 21.72243783838231234881411135137, 22.312237125909265622184882527737, 23.657897041689639937782735244692

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