Properties

Label 2-84-21.20-c11-0-9
Degree $2$
Conductor $84$
Sign $0.997 - 0.0634i$
Analytic cond. $64.5408$
Root an. cond. $8.03373$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−384. − 170. i)3-s + 2.40e3·5-s + (−3.94e4 + 2.06e4i)7-s + (1.18e5 + 1.31e5i)9-s − 6.69e5i·11-s − 5.28e5i·13-s + (−9.25e5 − 4.11e5i)15-s − 6.71e6·17-s + 3.29e6i·19-s + (1.86e7 − 1.18e6i)21-s + 5.21e7i·23-s − 4.30e7·25-s + (−2.31e7 − 7.08e7i)27-s + 8.39e7i·29-s − 2.41e8i·31-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)3-s + 0.344·5-s + (−0.886 + 0.463i)7-s + (0.669 + 0.742i)9-s − 1.25i·11-s − 0.394i·13-s + (−0.314 − 0.139i)15-s − 1.14·17-s + 0.305i·19-s + (0.997 − 0.0634i)21-s + 1.69i·23-s − 0.881·25-s + (−0.310 − 0.950i)27-s + 0.760i·29-s − 1.51i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0634i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.997 - 0.0634i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ 0.997 - 0.0634i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.952315 + 0.0302613i\)
\(L(\frac12)\) \(\approx\) \(0.952315 + 0.0302613i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (384. + 170. i)T \)
7 \( 1 + (3.94e4 - 2.06e4i)T \)
good5 \( 1 - 2.40e3T + 4.88e7T^{2} \)
11 \( 1 + 6.69e5iT - 2.85e11T^{2} \)
13 \( 1 + 5.28e5iT - 1.79e12T^{2} \)
17 \( 1 + 6.71e6T + 3.42e13T^{2} \)
19 \( 1 - 3.29e6iT - 1.16e14T^{2} \)
23 \( 1 - 5.21e7iT - 9.52e14T^{2} \)
29 \( 1 - 8.39e7iT - 1.22e16T^{2} \)
31 \( 1 + 2.41e8iT - 2.54e16T^{2} \)
37 \( 1 + 7.47e8T + 1.77e17T^{2} \)
41 \( 1 + 1.59e8T + 5.50e17T^{2} \)
43 \( 1 - 6.53e8T + 9.29e17T^{2} \)
47 \( 1 - 1.84e9T + 2.47e18T^{2} \)
53 \( 1 + 1.27e9iT - 9.26e18T^{2} \)
59 \( 1 - 8.68e9T + 3.01e19T^{2} \)
61 \( 1 - 1.10e10iT - 4.35e19T^{2} \)
67 \( 1 - 8.09e9T + 1.22e20T^{2} \)
71 \( 1 - 1.92e10iT - 2.31e20T^{2} \)
73 \( 1 + 2.94e10iT - 3.13e20T^{2} \)
79 \( 1 - 8.47e9T + 7.47e20T^{2} \)
83 \( 1 - 2.91e10T + 1.28e21T^{2} \)
89 \( 1 - 3.33e10T + 2.77e21T^{2} \)
97 \( 1 + 2.25e10iT - 7.15e21T^{2} \)
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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

−11.97041482456274102312906380429, −11.08781904424684527403061798395, −9.969216578482378607674245604441, −8.763207728836461231279081428202, −7.28533389959671822554310331244, −6.06499602118833542144866126830, −5.49384654981104243115725626517, −3.64732997688434565828165863431, −2.10162984198073050804279613451, −0.59944294791725693419379724449, 0.46242137289512159629184728612, 2.12965211154673731840636602745, 3.94945753037425875224591780302, 4.91519751263222895236823657589, 6.41683136675161343805645313188, 7.01916557794272697991051096368, 9.020095826760192831312882958932, 10.01447428569552173124743928683, 10.72517461712982043117861721721, 12.10232939445185405815974978421

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