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Properties

Label 2-2646-63.41-c1-0-37
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: $\chi_{2646} (881, \cdot )$
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

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

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