↑

Properties

Label 2-84-7.5-c8-0-9
Degree $2$
Conductor $84$
Sign $-0.784 + 0.620i$
Analytic cond. $34.2198$
Root an. cond. $5.84976$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (40.5 − 23.3i)3-s + (−225. − 129. i)5-s + (597. − 2.32e3i)7-s + (1.09e3 − 1.89e3i)9-s + (9.72e3 + 1.68e4i)11-s − 4.63e4i·13-s − 1.21e4·15-s + (−1.05e5 + 6.11e4i)17-s + (1.84e4 + 1.06e4i)19-s + (−3.01e4 − 1.08e5i)21-s + (1.52e5 − 2.63e5i)23-s + (−1.61e5 − 2.79e5i)25-s − 1.02e5i·27-s + 8.81e4·29-s + (−1.54e6 + 8.94e5i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−0.360 − 0.207i)5-s + (0.248 − 0.968i)7-s + (0.166 − 0.288i)9-s + (0.664 + 1.15i)11-s − 1.62i·13-s − 0.240·15-s + (−1.26 + 0.731i)17-s + (0.141 + 0.0817i)19-s + (−0.155 − 0.556i)21-s + (0.543 − 0.942i)23-s + (−0.413 − 0.716i)25-s − 0.192i·27-s + 0.124·29-s + (−1.67 + 0.968i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.784 + 0.620i$
Analytic conductor: \(34.2198\)
Root analytic conductor: \(5.84976\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :4),\ -0.784 + 0.620i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.482799 - 1.38900i\)
\(L(\frac12)\) \(\approx\) \(0.482799 - 1.38900i\)
\(L(5)\) 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 \( 1 + (-40.5 + 23.3i)T \)
7 \( 1 + (-597. + 2.32e3i)T \)
good5 \( 1 + (225. + 129. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-9.72e3 - 1.68e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 4.63e4iT - 8.15e8T^{2} \)
17 \( 1 + (1.05e5 - 6.11e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-1.84e4 - 1.06e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-1.52e5 + 2.63e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 8.81e4T + 5.00e11T^{2} \)
31 \( 1 + (1.54e6 - 8.94e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-7.55e5 + 1.30e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 1.88e5iT - 7.98e12T^{2} \)
43 \( 1 + 1.23e5T + 1.16e13T^{2} \)
47 \( 1 + (5.58e6 + 3.22e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (6.88e6 + 1.19e7i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-1.00e7 + 5.79e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (1.92e7 + 1.11e7i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-5.75e6 - 9.97e6i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 3.11e7T + 6.45e14T^{2} \)
73 \( 1 + (1.08e7 - 6.26e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-2.13e7 + 3.69e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 7.19e7iT - 2.25e15T^{2} \)
89 \( 1 + (-4.37e7 - 2.52e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 1.16e8iT - 7.83e15T^{2} \)
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

−12.53116556246860433445854319158, −11.01662590642836842183687422636, −10.04393729946154440533469860194, −8.653665781759691285157775445141, −7.65550955972344617972342380505, −6.64567521587070061151850704170, −4.74327474544968691891687053759, −3.60315003753011524173663828971, −1.85829606854534922642887153549, −0.39751667503129978225901329653, 1.77671294862478865232482246477, 3.21296731625857110812328213019, 4.53196829241092873604382096421, 6.08449616947018135551248912400, 7.42995645963449254760696296868, 8.975501514715742645654233307698, 9.226590517203486446656721859012, 11.38347189201966204028934708013, 11.47619062777725274454715680145, 13.25561803571545289966250140934

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