Properties

Label 1-1287-1287.358-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.126 + 0.991i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.994 + 0.104i)2-s + (0.978 − 0.207i)4-s + (0.406 + 0.913i)5-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (−0.5 − 0.866i)10-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (0.913 − 0.406i)17-s + (−0.207 + 0.978i)19-s + (0.587 + 0.809i)20-s − 23-s + (−0.669 + 0.743i)25-s + (0.994 + 0.104i)28-s + (−0.669 − 0.743i)29-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (0.978 − 0.207i)4-s + (0.406 + 0.913i)5-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (−0.5 − 0.866i)10-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (0.913 − 0.406i)17-s + (−0.207 + 0.978i)19-s + (0.587 + 0.809i)20-s − 23-s + (−0.669 + 0.743i)25-s + (0.994 + 0.104i)28-s + (−0.669 − 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.126 + 0.991i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (358, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.126 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8829479069 + 0.7775657775i\)
\(L(\frac12)\) \(\approx\) \(0.8829479069 + 0.7775657775i\)
\(L(1)\) \(\approx\) \(0.8191300572 + 0.2712257657i\)
\(L(1)\) \(\approx\) \(0.8191300572 + 0.2712257657i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.994 + 0.104i)T \)
5 \( 1 + (0.406 + 0.913i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (-0.207 + 0.978i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.669 - 0.743i)T \)
31 \( 1 + (0.994 - 0.104i)T \)
37 \( 1 + (0.207 + 0.978i)T \)
41 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.743 - 0.669i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.207 + 0.978i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.406 + 0.913i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (0.994 + 0.104i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−20.86957401797886522569120003225, −19.97636232179643928313191986377, −19.44174258863287052373013338461, −18.39024862861587894486381155695, −17.58978732901886036256354536632, −17.30579837581400457280306864743, −16.37445269672307460301471370366, −15.81279494873008584271230420301, −14.7162313103837703232351697051, −14.01668105058056272385730443322, −12.857253856214181980648192683534, −12.24786717655860298126015401245, −11.308913868685097879169257258639, −10.66814817762426288595535325509, −9.694297613321003560873766782938, −9.11711302902257268012015868490, −8.09692257880102600701264421676, −7.81350937282218461546741557093, −6.59096678937256751846476408268, −5.67418737679522839429906128289, −4.782619927405945563496443608108, −3.72565542876778599137157421020, −2.37236799832534325916599856680, −1.57800297483646700680933262535, −0.70553364530340130564790484920, 1.21246134073166922817315742547, 2.124178124774952389421431491140, 2.86448648016154056565655028586, 4.08729200017689157773284921661, 5.58061523434230324203518885389, 6.01152171413644499134090205805, 7.10319905257083970944872206586, 7.85673508826367738196759751150, 8.413641196118598526891949363381, 9.66800852231778262305507684070, 10.05273328378057477799455797642, 10.97850278302235083910499076508, 11.63636516976907989329878500205, 12.32626836110409215433441908688, 13.78525957317203173770683408818, 14.47051133559918026133571463738, 15.03477078876665358399400583344, 15.89238664539279370149146241240, 16.81621255168129116534140478639, 17.53974957919946002253834289520, 18.15205547822356893592066718042, 18.7829609635312901306271814944, 19.30886021616204440535418532708, 20.58632074161814322380753451424, 20.93165235318331178265280412531

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