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Properties

Label 2-84-28.27-c11-0-67
Degree $2$
Conductor $84$
Sign $0.997 + 0.0705i$
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  + (39.1 + 22.7i)2-s − 243·3-s + (1.01e3 + 1.78e3i)4-s + 1.13e4i·5-s + (−9.50e3 − 5.53e3i)6-s + (2.46e4 − 3.70e4i)7-s + (−946. + 9.26e4i)8-s + 5.90e4·9-s + (−2.57e5 + 4.43e5i)10-s − 8.36e5i·11-s + (−2.45e5 − 4.32e5i)12-s − 1.95e6i·13-s + (1.80e6 − 8.86e5i)14-s − 2.75e6i·15-s + (−2.14e6 + 3.60e6i)16-s − 3.85e6i·17-s + ⋯
L(s)  = 1  + (0.864 + 0.502i)2-s − 0.577·3-s + (0.494 + 0.869i)4-s + 1.62i·5-s + (−0.499 − 0.290i)6-s + (0.554 − 0.832i)7-s + (−0.0102 + 0.999i)8-s + 0.333·9-s + (−0.815 + 1.40i)10-s − 1.56i·11-s + (−0.285 − 0.501i)12-s − 1.45i·13-s + (0.897 − 0.440i)14-s − 0.935i·15-s + (−0.511 + 0.859i)16-s − 0.658i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0705i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.997 + 0.0705i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ 0.997 + 0.0705i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.82958 - 0.0999897i\)
\(L(\frac12)\) \(\approx\) \(2.82958 - 0.0999897i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-39.1 - 22.7i)T \)
3 \( 1 + 243T \)
7 \( 1 + (-2.46e4 + 3.70e4i)T \)
good5 \( 1 - 1.13e4iT - 4.88e7T^{2} \)
11 \( 1 + 8.36e5iT - 2.85e11T^{2} \)
13 \( 1 + 1.95e6iT - 1.79e12T^{2} \)
17 \( 1 + 3.85e6iT - 3.42e13T^{2} \)
19 \( 1 - 1.25e7T + 1.16e14T^{2} \)
23 \( 1 + 4.57e7iT - 9.52e14T^{2} \)
29 \( 1 + 7.66e7T + 1.22e16T^{2} \)
31 \( 1 - 1.44e8T + 2.54e16T^{2} \)
37 \( 1 + 3.19e8T + 1.77e17T^{2} \)
41 \( 1 - 7.67e8iT - 5.50e17T^{2} \)
43 \( 1 + 1.06e9iT - 9.29e17T^{2} \)
47 \( 1 - 3.18e8T + 2.47e18T^{2} \)
53 \( 1 - 4.49e9T + 9.26e18T^{2} \)
59 \( 1 + 4.58e9T + 3.01e19T^{2} \)
61 \( 1 + 6.89e9iT - 4.35e19T^{2} \)
67 \( 1 - 4.05e9iT - 1.22e20T^{2} \)
71 \( 1 + 1.54e10iT - 2.31e20T^{2} \)
73 \( 1 - 5.83e8iT - 3.13e20T^{2} \)
79 \( 1 + 5.69e9iT - 7.47e20T^{2} \)
83 \( 1 - 1.47e9T + 1.28e21T^{2} \)
89 \( 1 - 7.16e10iT - 2.77e21T^{2} \)
97 \( 1 + 5.54e10iT - 7.15e21T^{2} \)
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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

−11.87915583924646679554992808180, −10.97780810942344529462690070442, −10.44358496948851552622688216447, −8.107138525055458448204340524535, −7.20652086427393217071770302725, −6.23473513946755539029012073114, −5.20605057230261278916729040389, −3.59411996010616513447595947992, −2.79135638134728523634531244774, −0.57134175021559271668396647657, 1.32101429817485891900756152279, 1.89903840130461427465630727580, 4.15785974563222097036363160633, 4.92991586122477204097822230126, 5.71654202994975554172878167792, 7.31202238817213227762183418862, 9.050110324126671582257651933314, 9.830355784845422273424038273466, 11.59478555498938973390026418403, 12.00984572921777624030097073663

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