↑

Properties

Label 1-1287-1287.965-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.364 - 0.931i$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.978 + 0.207i)5-s + (−0.913 + 0.406i)7-s + (−0.809 + 0.587i)8-s + (0.5 − 0.866i)10-s + (0.104 + 0.994i)14-s + (0.309 + 0.951i)16-s + (0.669 − 0.743i)17-s + (−0.913 − 0.406i)19-s + (−0.669 − 0.743i)20-s + (0.5 + 0.866i)23-s + (0.913 + 0.406i)25-s + (0.978 + 0.207i)28-s + (−0.809 − 0.587i)29-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.978 + 0.207i)5-s + (−0.913 + 0.406i)7-s + (−0.809 + 0.587i)8-s + (0.5 − 0.866i)10-s + (0.104 + 0.994i)14-s + (0.309 + 0.951i)16-s + (0.669 − 0.743i)17-s + (−0.913 − 0.406i)19-s + (−0.669 − 0.743i)20-s + (0.5 + 0.866i)23-s + (0.913 + 0.406i)25-s + (0.978 + 0.207i)28-s + (−0.809 − 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.364 - 0.931i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (965, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.364 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.414822873 - 0.9660071617i\)
\(L(\frac12)\) \(\approx\) \(1.414822873 - 0.9660071617i\)
\(L(1)\) \(\approx\) \(1.107192519 - 0.5335953546i\)
\(L(1)\) \(\approx\) \(1.107192519 - 0.5335953546i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.978 + 0.207i)T \)
7 \( 1 + (-0.913 + 0.406i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.913 - 0.406i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.669 + 0.743i)T \)
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

−21.11907480208527725783715646288, −20.78828565316082466815418616290, −19.42260036038432393736319927297, −18.78073879667379994864597574515, −17.8765985120193537882997609494, −17.065960379845126162728432524383, −16.634636723045478830191192567604, −15.99991034515304815162360719580, −14.711079594923990018008302868912, −14.52283337437924708181367259394, −13.28919180489920192973377928641, −12.99488659178681660466854684746, −12.32241086162508214274130964668, −10.846093315115544462391029771269, −9.94009661086380398584326743480, −9.38394696335133131021665033727, −8.46617547356719662834791029600, −7.63313102213522637106485673564, −6.412656411112272083533804775160, −6.287361309470552044726930789716, −5.25942848163178958150996030960, −4.306813042761786556465324764714, −3.42742424931471672988139164376, −2.36134311189604252725226208885, −0.87387205642007534676503276827, 0.84123092544539045118522777778, 2.07148329116097669955376915000, 2.75648985532964952754280036187, 3.53414986466926406524164487750, 4.708001833819504990787052469461, 5.62937511287641488672964914580, 6.18024535597373976243701549283, 7.23995532451354895772008297251, 8.6621936130176838937159914525, 9.41265129404610800299827633164, 9.85300201937835221310093665843, 10.71131582153114681681898115611, 11.538698297135948558453619899805, 12.45706909730662748463597492866, 13.13031199358642254381085800844, 13.65570527218115788908979530363, 14.55392328260856729332704082456, 15.26668518217704434301593495448, 16.33506190868893837866666617341, 17.283126588792383645487308930, 17.96185263380288251814929634738, 18.81580739685379839300205039237, 19.268428167328707847559596608244, 20.142235638959391593758653001550, 21.13301786876370593968386780187

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