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Properties

Label 1-1287-1287.29-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.759 - 0.650i$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.669 − 0.743i)5-s + (0.104 + 0.994i)7-s + (−0.809 − 0.587i)8-s + (0.5 − 0.866i)10-s + (−0.913 + 0.406i)14-s + (0.309 − 0.951i)16-s + (−0.978 + 0.207i)17-s + (0.104 − 0.994i)19-s + (0.978 + 0.207i)20-s + (0.5 + 0.866i)23-s + (−0.104 + 0.994i)25-s + (−0.669 − 0.743i)28-s + (−0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.669 − 0.743i)5-s + (0.104 + 0.994i)7-s + (−0.809 − 0.587i)8-s + (0.5 − 0.866i)10-s + (−0.913 + 0.406i)14-s + (0.309 − 0.951i)16-s + (−0.978 + 0.207i)17-s + (0.104 − 0.994i)19-s + (0.978 + 0.207i)20-s + (0.5 + 0.866i)23-s + (−0.104 + 0.994i)25-s + (−0.669 − 0.743i)28-s + (−0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.759 - 0.650i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.759 - 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5606716378 - 0.2074911158i\)
\(L(\frac12)\) \(\approx\) \(0.5606716378 - 0.2074911158i\)
\(L(1)\) \(\approx\) \(0.7595768880 + 0.3177337907i\)
\(L(1)\) \(\approx\) \(0.7595768880 + 0.3177337907i\)

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 \( 1 + (-0.669 - 0.743i)T \)
7 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.104 - 0.994i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.978 - 0.207i)T \)
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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

−20.83335526197670836689328891418, −20.40338631905539306818332336921, −19.65280857047016708563494893736, −18.93118632312359240804116903722, −18.32094705098933708870552922862, −17.47961586788207813902636699719, −16.57389391871576922783252427456, −15.54090650205156910098387017345, −14.684642368217588806986605714848, −14.15730182107881026634633822981, −13.32443486592123314729528864707, −12.53946173770191421603165145541, −11.635572439061250635255812220093, −10.92928323326514367633287544634, −10.496168115773946814768034695539, −9.592825452948704098023047500042, −8.56903294375171851300412171024, −7.64355725346124850146178347046, −6.80044112429855691505842025586, −5.820660111380623511073579915498, −4.55122251116023382025518585875, −4.03654103065619862199178541521, −3.20171069089456063411035599838, −2.274144549078995156702492007583, −1.09024502175864033918063492852, 0.23736656430727828542561297981, 1.9073777927274864100826681621, 3.22857684209318939923054650074, 4.08412911218819173857426708768, 5.08427927987056634005224151828, 5.46771394552114156895657173144, 6.63742633813749123393805325075, 7.42180692991341987960877766112, 8.277380987392468935216827012706, 9.05577539065165379183706409531, 9.35646046662116786978951671634, 11.08887252592067000104817222859, 11.70664267999709740871457206464, 12.78416107040335063426772784037, 12.99507671582794805379225180396, 14.10643325438172155881065071830, 15.23996962880612460972025204742, 15.34205028501076619974917803627, 16.2074244837017802155566934370, 16.95350807291158472547737711570, 17.77536202521607751463132654690, 18.4399789529605033493573536360, 19.38197532670248023340959844977, 20.10834393166462797931251233628, 21.17174026448221456507255668987

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