↑

Properties

Label 2-84-28.27-c11-0-54
Degree $2$
Conductor $84$
Sign $-0.0644 + 0.997i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−32.5 − 31.4i)2-s − 243·3-s + (72.9 + 2.04e3i)4-s − 4.48e3i·5-s + (7.91e3 + 7.63e3i)6-s + (4.42e4 + 4.44e3i)7-s + (6.19e4 − 6.89e4i)8-s + 5.90e4·9-s + (−1.40e5 + 1.45e5i)10-s − 4.07e5i·11-s + (−1.77e4 − 4.97e5i)12-s − 1.73e6i·13-s + (−1.30e6 − 1.53e6i)14-s + 1.08e6i·15-s + (−4.18e6 + 2.98e5i)16-s + 5.93e6i·17-s + ⋯
L(s)  = 1  + (−0.719 − 0.694i)2-s − 0.577·3-s + (0.0356 + 0.999i)4-s − 0.641i·5-s + (0.415 + 0.400i)6-s + (0.994 + 0.0999i)7-s + (0.668 − 0.743i)8-s + 0.333·9-s + (−0.445 + 0.461i)10-s − 0.762i·11-s + (−0.0205 − 0.576i)12-s − 1.29i·13-s + (−0.646 − 0.762i)14-s + 0.370i·15-s + (−0.997 + 0.0711i)16-s + 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0644 + 0.997i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.0644 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.0644 + 0.997i$
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: \((2,\ 84,\ (\ :11/2),\ -0.0644 + 0.997i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.961973 - 1.02612i\)
\(L(\frac12)\) \(\approx\) \(0.961973 - 1.02612i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (32.5 + 31.4i)T \)
3 \( 1 + 243T \)
7 \( 1 + (-4.42e4 - 4.44e3i)T \)
good5 \( 1 + 4.48e3iT - 4.88e7T^{2} \)
11 \( 1 + 4.07e5iT - 2.85e11T^{2} \)
13 \( 1 + 1.73e6iT - 1.79e12T^{2} \)
17 \( 1 - 5.93e6iT - 3.42e13T^{2} \)
19 \( 1 - 1.52e7T + 1.16e14T^{2} \)
23 \( 1 - 2.31e6iT - 9.52e14T^{2} \)
29 \( 1 + 1.10e8T + 1.22e16T^{2} \)
31 \( 1 - 9.60e7T + 2.54e16T^{2} \)
37 \( 1 - 6.26e8T + 1.77e17T^{2} \)
41 \( 1 - 9.25e8iT - 5.50e17T^{2} \)
43 \( 1 - 1.26e9iT - 9.29e17T^{2} \)
47 \( 1 + 3.43e8T + 2.47e18T^{2} \)
53 \( 1 - 3.23e9T + 9.26e18T^{2} \)
59 \( 1 + 4.88e9T + 3.01e19T^{2} \)
61 \( 1 + 1.28e9iT - 4.35e19T^{2} \)
67 \( 1 + 1.47e10iT - 1.22e20T^{2} \)
71 \( 1 - 6.23e9iT - 2.31e20T^{2} \)
73 \( 1 - 7.64e9iT - 3.13e20T^{2} \)
79 \( 1 + 5.20e10iT - 7.47e20T^{2} \)
83 \( 1 + 3.89e10T + 1.28e21T^{2} \)
89 \( 1 - 6.32e10iT - 2.77e21T^{2} \)
97 \( 1 - 5.71e10iT - 7.15e21T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.50545526356114267080987267089, −10.83102531588936515882580963285, −9.652960280349970244063167682524, −8.374447939829152589624794867645, −7.69828924907713450027030687882, −5.81443272433371252120247587462, −4.61182362472166688885600251757, −3.10100904839884492873021903859, −1.39749446689824915189896909908, −0.68329257163833071453213186065, 0.919266980231118571625183531318, 2.15060149092968394856883265540, 4.47379739578523932103772692713, 5.50459844427951556567557949514, 6.96849415411103900125943770618, 7.45649125132217996428125480353, 9.033421365767701056510510499420, 10.02946192473071483538161414315, 11.18063617659087126286869608494, 11.81770370020122435558445232067

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