↑

Properties

Label 1-555-555.137-r0-0-0
Degree $1$
Conductor $555$
Sign $0.116 + 0.993i$
Analytic cond. $2.57740$
Root an. cond. $2.57740$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s + i·8-s − 11-s + (−0.866 + 0.5i)13-s + 14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.866 − 0.5i)22-s + i·23-s − 26-s + (0.866 + 0.5i)28-s + 29-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s + i·8-s − 11-s + (−0.866 + 0.5i)13-s + 14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.866 − 0.5i)22-s + i·23-s − 26-s + (0.866 + 0.5i)28-s + 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.116 + 0.993i$
Analytic conductor: \(2.57740\)
Root analytic conductor: \(2.57740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (0:\ ),\ 0.116 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.761214925 + 1.566131528i\)
\(L(\frac12)\) \(\approx\) \(1.761214925 + 1.566131528i\)
\(L(1)\) \(\approx\) \(1.599779676 + 0.7376000159i\)
\(L(1)\) \(\approx\) \(1.599779676 + 0.7376000159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + iT \)
29 \( 1 + T \)
31 \( 1 + T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 - iT \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + iT \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.03179722144904866544126678102, −22.289297386400109387422858227213, −21.37303087169769020435669320303, −20.858054283201475283653315177725, −20.0418945413114049726524297264, −19.05078016720114393820334963512, −18.27012526961244909893379215587, −17.40371798362281658103237057638, −16.01277629828400690281879324033, −15.414207115414967448981537527837, −14.47891750948903160930228988449, −13.87733468864767948917566203082, −12.708671124887956727389343273886, −12.14887850208058788298233175217, −11.20936711479384745062245465969, −10.39069478962719797301465600075, −9.511802278894384444111902959, −8.1596253440762065948269286768, −7.29503940924747864774058626927, −6.01585461422924299608183264820, −5.03219461500959420524275889640, −4.63321044639956237405925448011, −2.9209714329442731679063504713, −2.486299894883266323664566797277, −0.97885447089788364152318639677, 1.621997556737998911463010315121, 2.826848177082407869881067407213, 3.93376665601696542629398463551, 4.92373913819055105998241508444, 5.56314167471822727244721606346, 6.80860224968648811860813035822, 7.79513945962226198295728188863, 8.16497709357518240838231018009, 9.79710804528570826304777572386, 10.73911721464088847120607066712, 11.7947762665427658790487502760, 12.40393227421884802055245915261, 13.57256402979282360676333071401, 14.14776122291142712729011920616, 14.95150200361922939030398251840, 15.80488154380458719788519000936, 16.75882386492442749166227043325, 17.38994190307869566128905612217, 18.32970893612528541079002251270, 19.519942360893675533858172249244, 20.5056653435809876976885736365, 21.25773705319509354079719081048, 21.69776723822947787489057631770, 23.02310104297448536703626844897, 23.45644171815879163380482051973

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