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Properties

Label 2-2646-63.20-c1-0-12
Degree $2$
Conductor $2646$
Sign $-0.881 - 0.471i$
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 + (1.14 + 1.97i)5-s + 0.999i·8-s + 2.28i·10-s + (−0.946 − 0.546i)11-s + (−5.91 + 3.41i)13-s + (−0.5 + 0.866i)16-s + 6.71·17-s + 2.86i·19-s + (−1.14 + 1.97i)20-s + (−0.546 − 0.946i)22-s + (−3.38 + 1.95i)23-s + (−0.103 + 0.179i)25-s − 6.82·26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.510 + 0.883i)5-s + 0.353i·8-s + 0.721i·10-s + (−0.285 − 0.164i)11-s + (−1.64 + 0.947i)13-s + (−0.125 + 0.216i)16-s + 1.62·17-s + 0.656i·19-s + (−0.255 + 0.441i)20-s + (−0.116 − 0.201i)22-s + (−0.705 + 0.407i)23-s + (−0.0207 + 0.0358i)25-s − 1.33·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.881 - 0.471i$
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: \((2,\ 2646,\ (\ :1/2),\ -0.881 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.069651962\)
\(L(\frac12)\) \(\approx\) \(2.069651962\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.14 - 1.97i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.946 + 0.546i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.91 - 3.41i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.71T + 17T^{2} \)
19 \( 1 - 2.86iT - 19T^{2} \)
23 \( 1 + (3.38 - 1.95i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.59 + 0.923i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.75 + 1.01i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.15T + 37T^{2} \)
41 \( 1 + (-2.45 - 4.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.74 - 6.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.40 - 5.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.256iT - 53T^{2} \)
59 \( 1 + (0.971 + 1.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.15 + 0.665i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.54 + 4.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.233iT - 71T^{2} \)
73 \( 1 - 6.80iT - 73T^{2} \)
79 \( 1 + (-3.63 + 6.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.91 - 5.04i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 17.9T + 89T^{2} \)
97 \( 1 + (-4.13 - 2.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.384800484120339863451364089749, −8.030557785476705527512701325974, −7.61917749896268076988203682257, −6.78234828396683941322014660413, −6.12313867911826293140663590850, −5.35711840370524901317143314133, −4.56402946048058870244375622649, −3.48448019266485806401374425385, −2.72608437535631893860594390722, −1.77836044521053736229154723454, 0.50253077746775344252714415060, 1.78337961982718975599034807810, 2.73569606551167501326157631603, 3.64920408114020636254735939702, 4.93189763516016400386256331577, 5.15561101544376467406647824925, 5.87622322070542483800388760322, 7.06863274073756414111460040228, 7.68895309228299084651937218682, 8.604770801821396712258493464488

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