Properties

Degree $2$
Conductor $2646$
Sign $-0.0788 - 0.996i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 2·5-s + 0.999·8-s + (−1 − 1.73i)10-s − 11-s + (−3 − 5.19i)13-s + (−0.5 − 0.866i)16-s + (2.5 + 4.33i)17-s + (−3.5 + 6.06i)19-s + (−0.999 + 1.73i)20-s + (0.5 + 0.866i)22-s − 4·23-s − 25-s + (−3 + 5.19i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + 0.894·5-s + 0.353·8-s + (−0.316 − 0.547i)10-s − 0.301·11-s + (−0.832 − 1.44i)13-s + (−0.125 − 0.216i)16-s + (0.606 + 1.05i)17-s + (−0.802 + 1.39i)19-s + (−0.223 + 0.387i)20-s + (0.106 + 0.184i)22-s − 0.834·23-s − 0.200·25-s + (−0.588 + 1.01i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.0788 - 0.996i$
Motivic weight: \(1\)
Character: $\chi_{2646} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.0788 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6395124594\)
\(L(\frac12)\) \(\approx\) \(0.6395124594\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2T + 5T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8 - 13.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.217637976670619819793185243771, −8.168340322679005512462088057496, −7.937426045493439712861710692592, −6.80253076925883842234073445890, −5.68649778031052664928837546195, −5.43887385743767058580437677234, −4.09074320107573478716896028619, −3.25544249892602445848897696443, −2.23955877835577022395845030529, −1.43636455557231033643852398841, 0.21945234658843250227167295579, 1.88657746326511693864336009756, 2.55245853285665093412408415442, 4.11382886085275281724842610614, 4.88188944588736879478920717869, 5.66489699492468480610430555085, 6.43617715905182165911512521095, 7.13271091214475350525134201749, 7.76311535284028798622245006652, 8.810661984264330921149051843940

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