↑

Properties

Label 2-84-7.5-c8-0-6
Degree $2$
Conductor $84$
Sign $0.999 - 0.00269i$
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 + (656. + 379. i)5-s + (1.90e3 − 1.45e3i)7-s + (1.09e3 − 1.89e3i)9-s + (6.90e3 + 1.19e4i)11-s − 3.99e4i·13-s − 3.54e4·15-s + (3.30e4 − 1.91e4i)17-s + (−1.19e5 − 6.87e4i)19-s + (−4.31e4 + 1.03e5i)21-s + (−7.45e4 + 1.29e5i)23-s + (9.23e4 + 1.59e5i)25-s + 1.02e5i·27-s + 2.77e5·29-s + (1.48e6 − 8.59e5i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (1.05 + 0.606i)5-s + (0.794 − 0.607i)7-s + (0.166 − 0.288i)9-s + (0.471 + 0.817i)11-s − 1.39i·13-s − 0.700·15-s + (0.396 − 0.228i)17-s + (−0.913 − 0.527i)19-s + (−0.221 + 0.533i)21-s + (−0.266 + 0.461i)23-s + (0.236 + 0.409i)25-s + 0.192i·27-s + 0.392·29-s + (1.61 − 0.930i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00269i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.999 - 0.00269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.999 - 0.00269i$
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: \((2,\ 84,\ (\ :4),\ 0.999 - 0.00269i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.36011 + 0.00318283i\)
\(L(\frac12)\) \(\approx\) \(2.36011 + 0.00318283i\)
\(L(5)\) 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 \( 1 + (-1.90e3 + 1.45e3i)T \)
good5 \( 1 + (-656. - 379. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-6.90e3 - 1.19e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 3.99e4iT - 8.15e8T^{2} \)
17 \( 1 + (-3.30e4 + 1.91e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (1.19e5 + 6.87e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (7.45e4 - 1.29e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 2.77e5T + 5.00e11T^{2} \)
31 \( 1 + (-1.48e6 + 8.59e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-2.35e5 + 4.07e5i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 2.24e6iT - 7.98e12T^{2} \)
43 \( 1 - 4.59e6T + 1.16e13T^{2} \)
47 \( 1 + (-3.14e6 - 1.81e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (2.54e6 + 4.40e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-2.58e6 + 1.49e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-1.11e7 - 6.43e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (1.27e5 + 2.21e5i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 4.77e7T + 6.45e14T^{2} \)
73 \( 1 + (1.52e7 - 8.81e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (1.56e7 - 2.71e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 7.76e7iT - 2.25e15T^{2} \)
89 \( 1 + (7.90e7 + 4.56e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 2.05e7iT - 7.83e15T^{2} \)
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

−12.65592198471562844793550931953, −11.33195234557048613318396894376, −10.36755359290819142884618092518, −9.714763233573944243827278714595, −7.994080924824732657483921711059, −6.70240087747564083424534707820, −5.56512190366583269838663814199, −4.30437604666515545418464199035, −2.46928648382541512676245747072, −0.935080509550341958697490179083, 1.14313404752886134298403919417, 2.13791923552122512349137010687, 4.43172876058014956997173408852, 5.66484568921861392590656916904, 6.51036942353747216392976577314, 8.329063957764424853188723026695, 9.163337395332141321827401561639, 10.51429257463749924686797710535, 11.71236900915929095602921450597, 12.47608103466785494972026698625

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