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Properties

Label 2-84-21.20-c11-0-5
Degree $2$
Conductor $84$
Sign $-0.621 - 0.783i$
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 + (−1.16e7 − 1.46e7i)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 + (−0.621 − 0.783i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.621 - 0.783i$
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: \((2,\ 84,\ (\ :11/2),\ -0.621 - 0.783i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.587649 + 1.21606i\)
\(L(\frac12)\) \(\approx\) \(0.587649 + 1.21606i\)
\(L(\frac{13}{2})\) 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

−12.55873879051524164886511376036, −11.18088414986896374008025592062, −10.06256259696591694072717042878, −9.137277923748384760965994068081, −8.037449429264120793372812857602, −6.94582197255949374065198685229, −5.36284958008242153521404043080, −3.65652551770014867311239246875, −3.21330826898245191916618491257, −1.35277017664278430438092234190, 0.28816830036760012674858695635, 1.93832748915098145512374091669, 3.04478804845014786552966012853, 4.24425559460276971802805302040, 6.04391866375496191833468464890, 7.27292193873393181354097794042, 8.184514753380358927119174285990, 9.454177010850743016004025896863, 10.18734664966650660878443652193, 12.15823247692846164851309556141

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