Properties

Label 1-547-547.53-r0-0-0
Degree $1$
Conductor $547$
Sign $0.696 - 0.717i$
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  + (−0.944 + 0.327i)2-s + (−0.222 − 0.974i)3-s + (0.785 − 0.618i)4-s + (0.998 − 0.0460i)5-s + (0.529 + 0.848i)6-s + (0.983 + 0.183i)7-s + (−0.539 + 0.842i)8-s + (−0.900 + 0.433i)9-s + (−0.928 + 0.370i)10-s + (−0.200 + 0.979i)11-s + (−0.778 − 0.627i)12-s + (0.365 − 0.930i)13-s + (−0.988 + 0.149i)14-s + (−0.267 − 0.963i)15-s + (0.233 − 0.972i)16-s + (−0.991 + 0.126i)17-s + ⋯
L(s)  = 1  + (−0.944 + 0.327i)2-s + (−0.222 − 0.974i)3-s + (0.785 − 0.618i)4-s + (0.998 − 0.0460i)5-s + (0.529 + 0.848i)6-s + (0.983 + 0.183i)7-s + (−0.539 + 0.842i)8-s + (−0.900 + 0.433i)9-s + (−0.928 + 0.370i)10-s + (−0.200 + 0.979i)11-s + (−0.778 − 0.627i)12-s + (0.365 − 0.930i)13-s + (−0.988 + 0.149i)14-s + (−0.267 − 0.963i)15-s + (0.233 − 0.972i)16-s + (−0.991 + 0.126i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.696 - 0.717i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.696 - 0.717i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.000036957 - 0.4227524220i\)
\(L(\frac12)\) \(\approx\) \(1.000036957 - 0.4227524220i\)
\(L(1)\) \(\approx\) \(0.8513463476 - 0.1769668307i\)
\(L(1)\) \(\approx\) \(0.8513463476 - 0.1769668307i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.944 + 0.327i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (0.998 - 0.0460i)T \)
7 \( 1 + (0.983 + 0.183i)T \)
11 \( 1 + (-0.200 + 0.979i)T \)
13 \( 1 + (0.365 - 0.930i)T \)
17 \( 1 + (-0.991 + 0.126i)T \)
19 \( 1 + (0.211 - 0.977i)T \)
23 \( 1 + (0.924 - 0.381i)T \)
29 \( 1 + (0.770 + 0.636i)T \)
31 \( 1 + (-0.832 + 0.553i)T \)
37 \( 1 + (-0.667 - 0.744i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.300 - 0.953i)T \)
47 \( 1 + (0.692 - 0.721i)T \)
53 \( 1 + (0.968 + 0.250i)T \)
59 \( 1 + (-0.632 + 0.774i)T \)
61 \( 1 + (0.993 + 0.114i)T \)
67 \( 1 + (0.932 - 0.359i)T \)
71 \( 1 + (-0.819 + 0.572i)T \)
73 \( 1 + (-0.267 + 0.963i)T \)
79 \( 1 + (0.675 + 0.736i)T \)
83 \( 1 + (0.799 - 0.600i)T \)
89 \( 1 + (-0.596 - 0.802i)T \)
97 \( 1 + (0.641 - 0.767i)T \)
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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

−23.572254626846005836109505773034, −22.178940593553531889064018746733, −21.54794520820614563163295012294, −20.930200451022780462126659487231, −20.49200495730472823337059166841, −19.20130925317892302773886669875, −18.28857137796642021843972159943, −17.55543568227767018759690653696, −16.82954387497770626928806817058, −16.23611144282382970407182030093, −15.1671587728344172775815193792, −14.18941816712787701755112893025, −13.33489667629271069641730852728, −11.80687261752436563186256569496, −11.12688212981806833928009543330, −10.566889936463689767112879301804, −9.566483391549379754456682086006, −8.886773913260118937673517367575, −8.110696890350481995907395412334, −6.65946730279498387794667924065, −5.8019985814179500599565649746, −4.66256834064567322230289936424, −3.48441785054651881392746854945, −2.321843188779523937206624233801, −1.19763254178379303115081020953, 0.9428994208244991578131889051, 1.961902441945234231402639102266, 2.58540409255459112289542805584, 5.07080039563315390256156252300, 5.55839321270188347300399562210, 6.85922404665832054495100646106, 7.24947307485227111926210645998, 8.59270535557912262150356324631, 8.94197529850673033415721353204, 10.4840409829828814888568094299, 10.87828603176089337101575191133, 12.076978323237441031675997329657, 12.985229087555105336354777944665, 13.949565333671425251893451020810, 14.84413952622453086488795309035, 15.687740931250035191223950723997, 17.0820100491891156537744899549, 17.60999567033415098555811682318, 17.97951768863756259968841086948, 18.6827194632045699920390867121, 20.004649771469091866272039548804, 20.36999546233726140191548197895, 21.47743399373243945124824272000, 22.60692178614813414686080038773, 23.61250855259853599645113496249

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