↑

Properties

Label 1-1287-1287.1186-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.559 + 0.828i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.104 − 0.994i)5-s + (−0.978 − 0.207i)7-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (0.669 + 0.743i)14-s + (−0.809 + 0.587i)16-s + (0.913 + 0.406i)17-s + (−0.978 + 0.207i)19-s + (0.913 − 0.406i)20-s + (−0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.104 − 0.994i)28-s + (0.309 + 0.951i)29-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.104 − 0.994i)5-s + (−0.978 − 0.207i)7-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (0.669 + 0.743i)14-s + (−0.809 + 0.587i)16-s + (0.913 + 0.406i)17-s + (−0.978 + 0.207i)19-s + (0.913 − 0.406i)20-s + (−0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.104 − 0.994i)28-s + (0.309 + 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.559 + 0.828i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.559 + 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2819752573 + 0.1498948306i\)
\(L(\frac12)\) \(\approx\) \(0.2819752573 + 0.1498948306i\)
\(L(1)\) \(\approx\) \(0.5181155820 - 0.1754113912i\)
\(L(1)\) \(\approx\) \(0.5181155820 - 0.1754113912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.104 - 0.994i)T \)
7 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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

−20.790394120711749187148633886848, −19.71219484974320080091892778665, −19.101670174848613385354338079238, −18.78932305775252582653228396166, −17.80483701717977942798914074123, −17.16446210927106497854048362796, −16.26702303102132505197761138760, −15.52125725316880316790903125832, −15.07578416007163581688929028466, −14.0620207666824385504102360334, −13.480918810126458760246357584557, −12.12455617533029371899004006985, −11.47772694051001842699017360918, −10.20934395966104520529020871519, −10.16626526575568873934833826432, −9.12011938224200890410021609822, −8.18532824675545786952465874466, −7.37170907079458305478132110405, −6.57066576205346665090947444417, −6.09210098413838618740368937705, −5.052231798626712285528439853120, −3.65122339363148613109620134756, −2.78583051879645341753865404851, −1.7834265158277022368530940579, −0.19143113305728288782711980118, 0.985704792275075219112824892232, 1.965499781156500104459479142931, 3.14778033011978054116488437521, 3.90147376691780206446353677354, 4.8563557587146002907415750349, 6.16185012672749452222469427701, 6.88811082195105227250557155390, 8.14738405568071330948148796514, 8.43776971244851992046931700975, 9.52538495562191685971103901626, 10.02640395311899349872244846471, 10.83333079521044788506333800425, 11.96283465197800235397310859872, 12.54128758373663365081256542038, 13.00451391257511026160317770940, 14.003738855019767955873023681773, 15.27271992563293691102572288314, 16.14901827951604408334280918005, 16.66215772226826558916516521866, 17.17065243160245539513053580666, 18.17746599025433104340636418843, 19.06307834395327547469144954224, 19.547313950549235440190566075693, 20.24754405790423128030663845286, 20.98213272137812524423439001000

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