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Properties

Label 1-547-547.18-r1-0-0
Degree $1$
Conductor $547$
Sign $0.187 - 0.982i$
Analytic cond. $58.7833$
Root an. cond. $58.7833$
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  + (−0.895 − 0.444i)2-s + (0.900 − 0.433i)3-s + (0.605 + 0.795i)4-s + (−0.756 − 0.654i)5-s + (−0.999 − 0.0115i)6-s + (0.958 + 0.283i)7-s + (−0.188 − 0.982i)8-s + (0.623 − 0.781i)9-s + (0.386 + 0.922i)10-s + (0.692 + 0.721i)11-s + (0.890 + 0.454i)12-s + (0.955 − 0.294i)13-s + (−0.733 − 0.680i)14-s + (−0.965 − 0.261i)15-s + (−0.267 + 0.963i)16-s + (−0.924 − 0.381i)17-s + ⋯
L(s)  = 1  + (−0.895 − 0.444i)2-s + (0.900 − 0.433i)3-s + (0.605 + 0.795i)4-s + (−0.756 − 0.654i)5-s + (−0.999 − 0.0115i)6-s + (0.958 + 0.283i)7-s + (−0.188 − 0.982i)8-s + (0.623 − 0.781i)9-s + (0.386 + 0.922i)10-s + (0.692 + 0.721i)11-s + (0.890 + 0.454i)12-s + (0.955 − 0.294i)13-s + (−0.733 − 0.680i)14-s + (−0.965 − 0.261i)15-s + (−0.267 + 0.963i)16-s + (−0.924 − 0.381i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.187 - 0.982i$
Analytic conductor: \(58.7833\)
Root analytic conductor: \(58.7833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (1:\ ),\ 0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.679859620 - 1.389875263i\)
\(L(\frac12)\) \(\approx\) \(1.679859620 - 1.389875263i\)
\(L(1)\) \(\approx\) \(1.034070124 - 0.4622307838i\)
\(L(1)\) \(\approx\) \(1.034070124 - 0.4622307838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.895 - 0.444i)T \)
3 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (-0.756 - 0.654i)T \)
7 \( 1 + (0.958 + 0.283i)T \)
11 \( 1 + (0.692 + 0.721i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
17 \( 1 + (-0.924 - 0.381i)T \)
19 \( 1 + (0.586 - 0.809i)T \)
23 \( 1 + (0.976 - 0.216i)T \)
29 \( 1 + (-0.289 + 0.957i)T \)
31 \( 1 + (-0.322 - 0.946i)T \)
37 \( 1 + (0.439 + 0.898i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.717 + 0.696i)T \)
47 \( 1 + (-0.996 + 0.0804i)T \)
53 \( 1 + (0.709 - 0.705i)T \)
59 \( 1 + (0.919 - 0.391i)T \)
61 \( 1 + (-0.211 - 0.977i)T \)
67 \( 1 + (0.838 - 0.544i)T \)
71 \( 1 + (-0.0287 + 0.999i)T \)
73 \( 1 + (-0.965 + 0.261i)T \)
79 \( 1 + (-0.962 + 0.272i)T \)
83 \( 1 + (0.845 + 0.534i)T \)
89 \( 1 + (0.952 - 0.305i)T \)
97 \( 1 + (0.548 + 0.835i)T \)
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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

−23.599595764794618198469400327521, −22.63147290927637485774075344266, −21.38975310098020756867068601075, −20.68036913630156311193763270219, −19.795286270407076273216073385966, −19.17119718512422422389399769008, −18.46377695101397779354671900543, −17.54060265232527412004256343662, −16.40553767369666954084437909503, −15.78828802720815466158132670531, −14.887428807727822720964230837878, −14.359510329663556027731042984778, −13.55914174583939397536850063598, −11.71897039912014565143653736324, −10.98902988703879166163130633412, −10.45306934273214735037212295455, −9.05449080399051325319735578647, −8.58041186879024651817878304012, −7.7001479182684126800625262582, −6.98892821296611951952319611650, −5.76576096907450242709687030880, −4.31696307405007029103445997084, −3.48367777514905063401772017961, −2.147640017393877379378363646415, −1.03176264326394585729550584217, 0.839045015460355606367701698129, 1.5631252565083456153287491227, 2.718091233443671488605529129192, 3.83388258349996227763148758739, 4.773228633664914736880133772787, 6.66106803790405342048152702508, 7.482102129543585905509634566091, 8.28334217325145002225051144786, 8.9243399664196891144060894112, 9.54936163321539652303630176715, 11.19945636421162538877750804476, 11.53487473248921601573080558449, 12.704871625294955361281199424168, 13.25751095234303544769216680194, 14.70817518668707286093435433231, 15.378577079696384943510259833, 16.21841001134246693860048904433, 17.424699575040825831469690663782, 18.068864235408944669612403251776, 18.82687576311785171689226688970, 19.81641053871483362900921376086, 20.3237642804121409085611181627, 20.77799653042572434928902537628, 21.841230192387675337270977656161, 23.10011042572434979276549136064

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