↑

Properties

Label 2-69-23.22-c6-0-3
Degree $2$
Conductor $69$
Sign $-0.638 - 0.769i$
Analytic cond. $15.8737$
Root an. cond. $3.98418$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 10.8·2-s − 15.5·3-s + 54.8·4-s + 162. i·5-s − 169.·6-s − 219. i·7-s − 100.·8-s + 243·9-s + 1.77e3i·10-s + 1.60e3i·11-s − 854.·12-s − 3.92e3·13-s − 2.39e3i·14-s − 2.53e3i·15-s − 4.60e3·16-s + 6.86e3i·17-s + ⋯
L(s)  = 1  + 1.36·2-s − 0.577·3-s + 0.856·4-s + 1.29i·5-s − 0.786·6-s − 0.640i·7-s − 0.195·8-s + 0.333·9-s + 1.77i·10-s + 1.20i·11-s − 0.494·12-s − 1.78·13-s − 0.873i·14-s − 0.750i·15-s − 1.12·16-s + 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.638 - 0.769i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ -0.638 - 0.769i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.785329 + 1.67148i\)
\(L(\frac12)\) \(\approx\) \(0.785329 + 1.67148i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 15.5T \)
23 \( 1 + (-7.76e3 - 9.36e3i)T \)
good2 \( 1 - 10.8T + 64T^{2} \)
5 \( 1 - 162. iT - 1.56e4T^{2} \)
7 \( 1 + 219. iT - 1.17e5T^{2} \)
11 \( 1 - 1.60e3iT - 1.77e6T^{2} \)
13 \( 1 + 3.92e3T + 4.82e6T^{2} \)
17 \( 1 - 6.86e3iT - 2.41e7T^{2} \)
19 \( 1 + 1.47e3iT - 4.70e7T^{2} \)
29 \( 1 - 1.85e4T + 5.94e8T^{2} \)
31 \( 1 - 2.70e4T + 8.87e8T^{2} \)
37 \( 1 + 7.43e4iT - 2.56e9T^{2} \)
41 \( 1 + 5.33e4T + 4.75e9T^{2} \)
43 \( 1 + 6.86e4iT - 6.32e9T^{2} \)
47 \( 1 + 1.05e5T + 1.07e10T^{2} \)
53 \( 1 - 2.57e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.58e5T + 4.21e10T^{2} \)
61 \( 1 + 2.24e5iT - 5.15e10T^{2} \)
67 \( 1 - 5.30e5iT - 9.04e10T^{2} \)
71 \( 1 - 4.58e5T + 1.28e11T^{2} \)
73 \( 1 - 2.81e5T + 1.51e11T^{2} \)
79 \( 1 - 2.53e5iT - 2.43e11T^{2} \)
83 \( 1 - 1.47e5iT - 3.26e11T^{2} \)
89 \( 1 + 2.60e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.39e6iT - 8.32e11T^{2} \)
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

−14.02150155607270179614855297263, −12.76321358744542183432756122948, −12.00032752722671920271426852855, −10.75956033455756935034181039917, −9.852436672268762489958468424906, −7.31375285997809409492604285669, −6.62354325576752090170971197506, −5.13326601108489818016696746479, −3.97633315258912655047190391730, −2.44892132026896503127664473392, 0.47498184686253831374011900431, 2.82211976378789302507015953302, 4.84893853946283228031348220732, 5.11889126158153575448072687984, 6.55052296345125450617513129920, 8.461965227052426498686231936817, 9.675136350445846963547212183691, 11.62251542570565735933760489499, 12.13478165363825400003911891311, 13.01485485874370070877712771817

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