↑

Properties

Label 2-648-9.4-c3-0-2
Degree $2$
Conductor $648$
Sign $-0.939 - 0.342i$
Analytic cond. $38.2332$
Root an. cond. $6.18330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (5.08 − 8.81i)5-s + (14.5 + 25.2i)7-s + (−9.76 − 16.9i)11-s + (−30.0 + 52.1i)13-s − 102.·17-s − 113.·19-s + (7.41 − 12.8i)23-s + (10.7 + 18.5i)25-s + (−34.0 − 59.0i)29-s + (79.4 − 137. i)31-s + 296.·35-s − 75.8·37-s + (−182. + 315. i)41-s + (−145. − 252. i)43-s + (−227. − 393. i)47-s + ⋯
L(s)  = 1  + (0.455 − 0.788i)5-s + (0.787 + 1.36i)7-s + (−0.267 − 0.463i)11-s + (−0.641 + 1.11i)13-s − 1.46·17-s − 1.37·19-s + (0.0671 − 0.116i)23-s + (0.0856 + 0.148i)25-s + (−0.218 − 0.378i)29-s + (0.460 − 0.796i)31-s + 1.43·35-s − 0.336·37-s + (−0.693 + 1.20i)41-s + (−0.516 − 0.894i)43-s + (−0.705 − 1.22i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(38.2332\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :3/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4760312016\)
\(L(\frac12)\) \(\approx\) \(0.4760312016\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-5.08 + 8.81i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-14.5 - 25.2i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (9.76 + 16.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (30.0 - 52.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 102.T + 4.91e3T^{2} \)
19 \( 1 + 113.T + 6.85e3T^{2} \)
23 \( 1 + (-7.41 + 12.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (34.0 + 59.0i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-79.4 + 137. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 75.8T + 5.06e4T^{2} \)
41 \( 1 + (182. - 315. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (145. + 252. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (227. + 393. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 560.T + 1.48e5T^{2} \)
59 \( 1 + (179. - 310. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (191. + 331. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (368. - 638. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 360.T + 3.57e5T^{2} \)
73 \( 1 + 1.01e3T + 3.89e5T^{2} \)
79 \( 1 + (-296. - 512. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (115. + 200. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.32e3T + 7.04e5T^{2} \)
97 \( 1 + (-236. - 409. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−10.57246386328604115504043953227, −9.401085239033129941354317949925, −8.738863104151952087215534960752, −8.343975224876392069290409177223, −6.87493077085931540941451257785, −5.94171062991718319839567348816, −5.02489127424313060959225796348, −4.32919789987927628988843766250, −2.43486581361200335138747293271, −1.79522435663794948588792143268, 0.12109995951226200537184716676, 1.75840110417745940653581482331, 2.87889496293838905717336712905, 4.25463472175053198906267642229, 4.99380943344535363350738195824, 6.39610938623401255562133268234, 7.08870989300979748698456039599, 7.87610583135467232056524539459, 8.847003557717669643380778393161, 10.22547325611073540149417902980

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