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Properties

Label 1-547-547.296-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.791 - 0.610i$
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  + (0.799 − 0.600i)2-s + 3-s + (0.278 − 0.960i)4-s + (−0.632 − 0.774i)5-s + (0.799 − 0.600i)6-s + (−0.919 + 0.391i)7-s + (−0.354 − 0.935i)8-s + 9-s + (−0.970 − 0.239i)10-s + (0.428 − 0.903i)11-s + (0.278 − 0.960i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.632 − 0.774i)15-s + (−0.845 − 0.534i)16-s + (−0.996 − 0.0804i)17-s + ⋯
L(s)  = 1  + (0.799 − 0.600i)2-s + 3-s + (0.278 − 0.960i)4-s + (−0.632 − 0.774i)5-s + (0.799 − 0.600i)6-s + (−0.919 + 0.391i)7-s + (−0.354 − 0.935i)8-s + 9-s + (−0.970 − 0.239i)10-s + (0.428 − 0.903i)11-s + (0.278 − 0.960i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.632 − 0.774i)15-s + (−0.845 − 0.534i)16-s + (−0.996 − 0.0804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.791 - 0.610i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.791 - 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7078811326 - 2.077223543i\)
\(L(\frac12)\) \(\approx\) \(0.7078811326 - 2.077223543i\)
\(L(1)\) \(\approx\) \(1.338132352 - 1.094956589i\)
\(L(1)\) \(\approx\) \(1.338132352 - 1.094956589i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.799 - 0.600i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.632 - 0.774i)T \)
7 \( 1 + (-0.919 + 0.391i)T \)
11 \( 1 + (0.428 - 0.903i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.996 - 0.0804i)T \)
19 \( 1 + (-0.845 + 0.534i)T \)
23 \( 1 + (0.948 - 0.316i)T \)
29 \( 1 + (-0.748 + 0.663i)T \)
31 \( 1 + (0.885 - 0.464i)T \)
37 \( 1 + (0.948 + 0.316i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.692 + 0.721i)T \)
47 \( 1 + (-0.200 - 0.979i)T \)
53 \( 1 + (0.987 - 0.160i)T \)
59 \( 1 + (-0.845 - 0.534i)T \)
61 \( 1 + (0.799 - 0.600i)T \)
67 \( 1 + (0.692 + 0.721i)T \)
71 \( 1 + (-0.0402 - 0.999i)T \)
73 \( 1 + (-0.632 + 0.774i)T \)
79 \( 1 + (-0.970 + 0.239i)T \)
83 \( 1 + (0.987 - 0.160i)T \)
89 \( 1 + (0.120 - 0.992i)T \)
97 \( 1 + (0.428 + 0.903i)T \)
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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

−23.62845367126794796691329897347, −22.9432374002969231503675560106, −22.13899909447071041677295079451, −21.42169318800450366526722181588, −20.29595662832105296819116875540, −19.599130354123402510173551427256, −18.99645900502287310846346944176, −17.691461540685835664158151568406, −16.74159748680345092463958104620, −15.69017467055784748879780620660, −15.13746007064039571613005880167, −14.52637392807677416914049609115, −13.55882582284673140994633203674, −12.90475070155507552882246470112, −11.94718081649647775568354589113, −10.868392102325728545803098737928, −9.62733203090241940276908880162, −8.79444654579909844108004198456, −7.5684764709531573193229728236, −6.950761612061008901217699852130, −6.446002219236105056382533001470, −4.43780739042139495432902810015, −4.12416379677549219304949739972, −2.97094889379072977314683520131, −2.2299979497971405660761909799, 0.75263057650968402717693645788, 2.22680947494353963687216368451, 3.17925291982867203489616993306, 3.899408868269729012190003790, 4.87995305639347896497396214344, 6.061779067164703517585057027303, 7.107257765251429330494638294511, 8.44047413985349791566627717622, 9.07819876261997206801786093082, 10.01271587935371213830795914101, 11.08921088276271914156275575995, 12.17205543447461065765602861040, 12.99982809404449117303250598802, 13.27131811985075506746030895089, 14.56931035844347690589367513036, 15.26846794197999760267254406663, 15.91312405943673993381971004018, 16.8811670990454809302506464544, 18.61857285824750352884327493771, 19.19412871052835908739185503244, 19.843803859090308196421606723248, 20.40914493016262695196866578126, 21.33218003184582807304183704953, 22.11799776545871779532160857495, 22.92557175185237239793341159400

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