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Properties

Label 2-1287-13.12-c1-0-36
Degree $2$
Conductor $1287$
Sign $0.920 - 0.390i$
Analytic cond. $10.2767$
Root an. cond. $3.20573$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.15i·2-s − 2.65·4-s − 0.710i·5-s − 2.30i·7-s − 1.41i·8-s + 1.53·10-s + i·11-s + (−1.40 − 3.31i)13-s + 4.98·14-s − 2.26·16-s − 6.68·17-s − 0.242i·19-s + 1.88i·20-s − 2.15·22-s + 9.53·23-s + ⋯
L(s)  = 1  + 1.52i·2-s − 1.32·4-s − 0.317i·5-s − 0.872i·7-s − 0.499i·8-s + 0.484·10-s + 0.301i·11-s + (−0.390 − 0.920i)13-s + 1.33·14-s − 0.565·16-s − 1.62·17-s − 0.0556i·19-s + 0.421i·20-s − 0.459·22-s + 1.98·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.920 - 0.390i$
Analytic conductor: \(10.2767\)
Root analytic conductor: \(3.20573\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :1/2),\ 0.920 - 0.390i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.273848470\)
\(L(\frac12)\) \(\approx\) \(1.273848470\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 - iT \)
13 \( 1 + (1.40 + 3.31i)T \)
good2 \( 1 - 2.15iT - 2T^{2} \)
5 \( 1 + 0.710iT - 5T^{2} \)
7 \( 1 + 2.30iT - 7T^{2} \)
17 \( 1 + 6.68T + 17T^{2} \)
19 \( 1 + 0.242iT - 19T^{2} \)
23 \( 1 - 9.53T + 23T^{2} \)
29 \( 1 - 2.95T + 29T^{2} \)
31 \( 1 + 4.02iT - 31T^{2} \)
37 \( 1 + 3.42iT - 37T^{2} \)
41 \( 1 + 9.46iT - 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 + 12.9iT - 47T^{2} \)
53 \( 1 + 6.78T + 53T^{2} \)
59 \( 1 - 7.81iT - 59T^{2} \)
61 \( 1 - 0.910T + 61T^{2} \)
67 \( 1 - 9.32iT - 67T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 + 8.51iT - 73T^{2} \)
79 \( 1 - 1.82T + 79T^{2} \)
83 \( 1 + 4.64iT - 83T^{2} \)
89 \( 1 - 14.7iT - 89T^{2} \)
97 \( 1 - 1.86iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.212020922748677266345039595894, −8.845913538203780670832386056789, −7.84937607297953050847242579673, −7.11826242820033483625618612880, −6.72153401934613266693842942728, −5.55021696321298729727745266295, −4.85423499262957780345844951936, −4.08935646312488815213970150973, −2.54422933298603432282104394021, −0.58033127968589020879443690454, 1.28768441255568059418671920364, 2.55871103429655513070102818247, 2.96360064121659312123667778739, 4.35160280666913937441939530457, 4.95487592914166085218541468706, 6.39919334205568928463879628337, 6.99040491397785619509090360906, 8.457064994281013243556524901565, 9.152284964847888462385599283955, 9.532699282024679507312867770862

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