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Properties

Label 2-2646-63.20-c1-0-10
Degree $2$
Conductor $2646$
Sign $-0.991 - 0.130i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (1.35 + 2.33i)5-s + 0.999i·8-s + 2.70i·10-s + (0.205 + 0.118i)11-s + (−2.31 + 1.33i)13-s + (−0.5 + 0.866i)16-s − 5.86·17-s + 1.15i·19-s + (−1.35 + 2.33i)20-s + (0.118 + 0.205i)22-s + (−7.02 + 4.05i)23-s + (−1.14 + 1.98i)25-s − 2.67·26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.603 + 1.04i)5-s + 0.353i·8-s + 0.853i·10-s + (0.0621 + 0.0358i)11-s + (−0.641 + 0.370i)13-s + (−0.125 + 0.216i)16-s − 1.42·17-s + 0.264i·19-s + (−0.301 + 0.522i)20-s + (0.0253 + 0.0439i)22-s + (−1.46 + 0.845i)23-s + (−0.229 + 0.397i)25-s − 0.524·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.991 - 0.130i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.708397376\)
\(L(\frac12)\) \(\approx\) \(1.708397376\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1.35 - 2.33i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.205 - 0.118i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.31 - 1.33i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.86T + 17T^{2} \)
19 \( 1 - 1.15iT - 19T^{2} \)
23 \( 1 + (7.02 - 4.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.88 + 5.13i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.21 - 3.01i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.68T + 37T^{2} \)
41 \( 1 + (-3.81 - 6.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.69 + 4.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.221 - 0.383i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.219iT - 53T^{2} \)
59 \( 1 + (-0.983 - 1.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.8 - 6.27i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.48 - 7.76i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.24iT - 71T^{2} \)
73 \( 1 - 7.25iT - 73T^{2} \)
79 \( 1 + (-5.43 + 9.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.762 + 1.32i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 + (-1.37 - 0.795i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.410219601862241537244515241863, −8.285041993327312445850724201324, −7.42701315115904140077112345766, −6.90353164173719634287652188424, −6.06815989184725578895269749039, −5.60262952537868511183766861350, −4.38316584833451383576265035363, −3.76784014072112483278574250341, −2.53477512670747447207736222024, −2.01829714711857107566878206675, 0.39489255962955160571110399887, 1.83573266925211743740850978111, 2.46799872780094276176387563871, 3.85964693604926731292693034233, 4.51950730307621505478490446620, 5.33324304987444767423766572639, 5.91549470446777031153893622088, 6.81938679630081203174092338286, 7.74607121967323758439141671436, 8.646958082882134237150209404078

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