Properties

Degree $2$
Conductor $2646$
Sign $0.841 + 0.540i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + (−1.5 + 2.59i)5-s + 8-s + (−1.5 + 2.59i)10-s + (−3 − 5.19i)11-s + (−1 − 1.73i)13-s + 16-s + (3 − 5.19i)17-s + (3.5 + 6.06i)19-s + (−1.5 + 2.59i)20-s + (−3 − 5.19i)22-s + (1.5 − 2.59i)23-s + (−2 − 3.46i)25-s + (−1 − 1.73i)26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.670 + 1.16i)5-s + 0.353·8-s + (−0.474 + 0.821i)10-s + (−0.904 − 1.56i)11-s + (−0.277 − 0.480i)13-s + 0.250·16-s + (0.727 − 1.26i)17-s + (0.802 + 1.39i)19-s + (−0.335 + 0.580i)20-s + (−0.639 − 1.10i)22-s + (0.312 − 0.541i)23-s + (−0.400 − 0.692i)25-s + (−0.196 − 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.841 + 0.540i$
Motivic weight: \(1\)
Character: $\chi_{2646} (2125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.841 + 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.247104561\)
\(L(\frac12)\) \(\approx\) \(2.247104561\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.482083733865472863739072583068, −7.81483241923227660534897359869, −7.34813278557772562998255303352, −6.37983515279038426289165636690, −5.66977308422121014614917719817, −4.97916902072445454831474191262, −3.69505231734168906113720480524, −3.15387774915306592228537812419, −2.56211467694364357154000062474, −0.65171701235582803956063941108, 1.14394034643935806918428788045, 2.27440605358588165145481548380, 3.41390121704826515959585786245, 4.41042664073731870777256956837, 4.90697133009652670068329192723, 5.45803972849861360057363719879, 6.74491255579741140628666536701, 7.38142009381650076942374024511, 8.042890366432348679201748384642, 8.858950556923864377819119323278

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