↑

Properties

Label 2-12e2-48.35-c3-0-6
Degree $2$
Conductor $144$
Sign $-0.963 - 0.267i$
Analytic cond. $8.49627$
Root an. cond. $2.91483$
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  + (1.05 + 2.62i)2-s + (−5.78 + 5.52i)4-s + (11.8 + 11.8i)5-s − 12.7·7-s + (−20.5 − 9.39i)8-s + (−18.6 + 43.5i)10-s + (−15.5 + 15.5i)11-s + (42.1 + 42.1i)13-s + (−13.4 − 33.4i)14-s + (3.04 − 63.9i)16-s − 76.6i·17-s + (−108. + 108. i)19-s + (−134. − 3.18i)20-s + (−57.1 − 24.4i)22-s − 52.0i·23-s + ⋯
L(s)  = 1  + (0.371 + 0.928i)2-s + (−0.723 + 0.690i)4-s + (1.05 + 1.05i)5-s − 0.688·7-s + (−0.909 − 0.415i)8-s + (−0.589 + 1.37i)10-s + (−0.425 + 0.425i)11-s + (0.900 + 0.900i)13-s + (−0.255 − 0.639i)14-s + (0.0475 − 0.998i)16-s − 1.09i·17-s + (−1.31 + 1.31i)19-s + (−1.49 − 0.0356i)20-s + (−0.553 − 0.237i)22-s − 0.471i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.963 - 0.267i$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ -0.963 - 0.267i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.232928 + 1.71286i\)
\(L(\frac12)\) \(\approx\) \(0.232928 + 1.71286i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.05 - 2.62i)T \)
3 \( 1 \)
good5 \( 1 + (-11.8 - 11.8i)T + 125iT^{2} \)
7 \( 1 + 12.7T + 343T^{2} \)
11 \( 1 + (15.5 - 15.5i)T - 1.33e3iT^{2} \)
13 \( 1 + (-42.1 - 42.1i)T + 2.19e3iT^{2} \)
17 \( 1 + 76.6iT - 4.91e3T^{2} \)
19 \( 1 + (108. - 108. i)T - 6.85e3iT^{2} \)
23 \( 1 + 52.0iT - 1.21e4T^{2} \)
29 \( 1 + (89.7 - 89.7i)T - 2.43e4iT^{2} \)
31 \( 1 + 12.9iT - 2.97e4T^{2} \)
37 \( 1 + (-115. + 115. i)T - 5.06e4iT^{2} \)
41 \( 1 - 307.T + 6.89e4T^{2} \)
43 \( 1 + (-342. - 342. i)T + 7.95e4iT^{2} \)
47 \( 1 - 357.T + 1.03e5T^{2} \)
53 \( 1 + (-450. - 450. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-395. + 395. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-220. - 220. i)T + 2.26e5iT^{2} \)
67 \( 1 + (243. - 243. i)T - 3.00e5iT^{2} \)
71 \( 1 + 414. iT - 3.57e5T^{2} \)
73 \( 1 + 91.5iT - 3.89e5T^{2} \)
79 \( 1 - 236. iT - 4.93e5T^{2} \)
83 \( 1 + (204. + 204. i)T + 5.71e5iT^{2} \)
89 \( 1 - 688.T + 7.04e5T^{2} \)
97 \( 1 - 968.T + 9.12e5T^{2} \)
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

−13.32927197628525418756062856853, −12.48329976676314721109758246373, −10.93563960859758208194123864008, −9.845527055990038840172671292848, −8.965826053956115469843633501923, −7.43736927917728241104659892987, −6.45617095028589669745066918304, −5.81946032664328799819726993464, −4.08752715684013821090373274932, −2.56518886796934323412651161926, 0.75936014432660951842610355686, 2.36812613103534826304277681731, 3.98080082285546105621645995655, 5.44864455512322673374590710559, 6.14169319310472573519596827883, 8.462990642918642179690157351083, 9.178153073334265846637760027304, 10.24568446318673808715979991371, 11.02877654978315746546250652031, 12.53383318744617815664952833186

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