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Properties

Label 1-547-547.318-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.246 + 0.969i$
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  + (0.0976 + 0.995i)2-s + (−0.900 − 0.433i)3-s + (−0.980 + 0.194i)4-s + (0.973 − 0.228i)5-s + (0.343 − 0.939i)6-s + (0.605 + 0.795i)7-s + (−0.289 − 0.957i)8-s + (0.623 + 0.781i)9-s + (0.322 + 0.946i)10-s + (−0.845 + 0.534i)11-s + (0.968 + 0.250i)12-s + (0.955 + 0.294i)13-s + (−0.733 + 0.680i)14-s + (−0.976 − 0.216i)15-s + (0.924 − 0.381i)16-s + (−0.806 + 0.591i)17-s + ⋯
L(s)  = 1  + (0.0976 + 0.995i)2-s + (−0.900 − 0.433i)3-s + (−0.980 + 0.194i)4-s + (0.973 − 0.228i)5-s + (0.343 − 0.939i)6-s + (0.605 + 0.795i)7-s + (−0.289 − 0.957i)8-s + (0.623 + 0.781i)9-s + (0.322 + 0.946i)10-s + (−0.845 + 0.534i)11-s + (0.968 + 0.250i)12-s + (0.955 + 0.294i)13-s + (−0.733 + 0.680i)14-s + (−0.976 − 0.216i)15-s + (0.924 − 0.381i)16-s + (−0.806 + 0.591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.246 + 0.969i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.246 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7082699543 + 0.9114115404i\)
\(L(\frac12)\) \(\approx\) \(0.7082699543 + 0.9114115404i\)
\(L(1)\) \(\approx\) \(0.8280877496 + 0.4851116934i\)
\(L(1)\) \(\approx\) \(0.8280877496 + 0.4851116934i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.0976 + 0.995i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (0.973 - 0.228i)T \)
7 \( 1 + (0.605 + 0.795i)T \)
11 \( 1 + (-0.845 + 0.534i)T \)
13 \( 1 + (0.955 + 0.294i)T \)
17 \( 1 + (-0.806 + 0.591i)T \)
19 \( 1 + (0.874 - 0.484i)T \)
23 \( 1 + (-0.376 - 0.926i)T \)
29 \( 1 + (-0.952 - 0.305i)T \)
31 \( 1 + (0.978 + 0.205i)T \)
37 \( 1 + (0.490 + 0.871i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.998 - 0.0460i)T \)
47 \( 1 + (-0.632 + 0.774i)T \)
53 \( 1 + (0.300 + 0.953i)T \)
59 \( 1 + (0.278 - 0.960i)T \)
61 \( 1 + (0.838 + 0.544i)T \)
67 \( 1 + (-0.267 - 0.963i)T \)
71 \( 1 + (0.995 + 0.0919i)T \)
73 \( 1 + (-0.976 + 0.216i)T \)
79 \( 1 + (-0.539 - 0.842i)T \)
83 \( 1 + (-0.996 + 0.0804i)T \)
89 \( 1 + (0.0517 + 0.998i)T \)
97 \( 1 + (-0.332 + 0.942i)T \)
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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

−22.83603548361735110016842455128, −22.290320189769629050814209405622, −21.2459772108850557068865826423, −20.9240457709926242698163021441, −20.16440888617713265603764417741, −18.69625821501740647990840727678, −17.98012816166260093237851177777, −17.63880058828848394419961129633, −16.57874987939238598812019002614, −15.58696133978695393387233588380, −14.30024018689919932219167140763, −13.52663868238167226759448272319, −12.97807981342866806219469015192, −11.54400478698762741649538208107, −11.088845553248406407440504809895, −10.30957596963454115870115953504, −9.6853440198144856724586951878, −8.55779473946774108686290963990, −7.220111748084267717159027967289, −5.7588303613162492926958420444, −5.36222215762644808156655981492, −4.200086183894852427555153863481, −3.2391464967665820533012491128, −1.841405902726580491924120182954, −0.772416448743416366521321656617, 1.260460000275279960552923475832, 2.45584065778972959116283439721, 4.48657981705612953564332774974, 5.11585644578433908293350788561, 6.00740269600941350075576448292, 6.52572241209925009947620975869, 7.77139669700654535236277438812, 8.59440534811218411860775750498, 9.611688736293366210434826176716, 10.62544920543758305668479944441, 11.72471682533724122896289642012, 12.83848890863078106512356147852, 13.282358307773862548430164703569, 14.22896863293197197156019870827, 15.382352553156383346685557419017, 16.01629735744615075975583830154, 17.001383576120793453182247336372, 17.764977988509598664750807600, 18.20364281388118744267330771326, 18.82030100975841692107147986463, 20.57160856545799736273584591065, 21.447226901965249942146915359395, 22.131555925397339932611417770919, 22.835865766055267561499551605879, 23.933257613766551235638899566724

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