↑

Properties

Label 2-648-9.7-c3-0-24
Degree $2$
Conductor $648$
Sign $0.342 + 0.939i$
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  + (3.16 + 5.49i)5-s + (−9.59 + 16.6i)7-s + (20.6 − 35.7i)11-s + (−19.6 − 34.0i)13-s − 59.9·17-s − 41.2·19-s + (−26.9 − 46.5i)23-s + (42.4 − 73.4i)25-s + (−32.6 + 56.5i)29-s + (−32.3 − 56.0i)31-s − 121.·35-s + 293.·37-s + (103. + 179. i)41-s + (188. − 327. i)43-s + (161. − 280. i)47-s + ⋯
L(s)  = 1  + (0.283 + 0.491i)5-s + (−0.518 + 0.897i)7-s + (0.565 − 0.979i)11-s + (−0.418 − 0.725i)13-s − 0.855·17-s − 0.498·19-s + (−0.243 − 0.422i)23-s + (0.339 − 0.587i)25-s + (−0.208 + 0.361i)29-s + (−0.187 − 0.324i)31-s − 0.587·35-s + 1.30·37-s + (0.394 + 0.683i)41-s + (0.670 − 1.16i)43-s + (0.502 − 0.870i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(38.2332\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :3/2),\ 0.342 + 0.939i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.348982412\)
\(L(\frac12)\) \(\approx\) \(1.348982412\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.16 - 5.49i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (9.59 - 16.6i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-20.6 + 35.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (19.6 + 34.0i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 59.9T + 4.91e3T^{2} \)
19 \( 1 + 41.2T + 6.85e3T^{2} \)
23 \( 1 + (26.9 + 46.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (32.6 - 56.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (32.3 + 56.0i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 293.T + 5.06e4T^{2} \)
41 \( 1 + (-103. - 179. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-188. + 327. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-161. + 280. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 340.T + 1.48e5T^{2} \)
59 \( 1 + (-203. - 352. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-378. + 655. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (422. + 731. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 859.T + 3.57e5T^{2} \)
73 \( 1 + 789.T + 3.89e5T^{2} \)
79 \( 1 + (-108. + 188. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-50.5 + 87.6i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.24e3T + 7.04e5T^{2} \)
97 \( 1 + (-873. + 1.51e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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

−10.02617416805946482721773974546, −9.014411998390083098042149914618, −8.479724571588978212986852178151, −7.22470271416588073190995303369, −6.21489493538424486752914662019, −5.74552402561686566413090943414, −4.34519711596556680075492418368, −3.07400714080340161546492485584, −2.26760395480274083293187966727, −0.41251172990742218295912755386, 1.16184821966741417780784755549, 2.39234453044075341383873675516, 4.03035380923194546497573252797, 4.53740268623135584581224603169, 5.89077653423021075932311596368, 6.91003745661295324587751113825, 7.45190519162321763990769625805, 8.829631227156533928826026560739, 9.461544539557104458498641206721, 10.16889175208465563520988275085

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