Properties

Degree $2$
Conductor $144$
Sign $-0.799 + 0.600i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.125 + 1.99i)2-s + (−3.96 + 0.500i)4-s + (−3.32 + 3.32i)5-s − 4.04·7-s + (−1.49 − 7.85i)8-s + (−7.05 − 6.22i)10-s + (−6.82 − 6.82i)11-s + (4.29 + 4.29i)13-s + (−0.506 − 8.06i)14-s + (15.4 − 3.97i)16-s − 30.1·17-s + (−19.7 + 19.7i)19-s + (11.5 − 14.8i)20-s + (12.7 − 14.4i)22-s + 28.2·23-s + ⋯
L(s)  = 1  + (0.0626 + 0.998i)2-s + (−0.992 + 0.125i)4-s + (−0.665 + 0.665i)5-s − 0.577·7-s + (−0.187 − 0.982i)8-s + (−0.705 − 0.622i)10-s + (−0.620 − 0.620i)11-s + (0.330 + 0.330i)13-s + (−0.0361 − 0.576i)14-s + (0.968 − 0.248i)16-s − 1.77·17-s + (−1.03 + 1.03i)19-s + (0.576 − 0.743i)20-s + (0.580 − 0.658i)22-s + 1.22·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.799 + 0.600i$
Motivic weight: \(2\)
Character: $\chi_{144} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.799 + 0.600i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.134608 - 0.403772i\)
\(L(\frac12)\) \(\approx\) \(0.134608 - 0.403772i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.125 - 1.99i)T \)
3 \( 1 \)
good5 \( 1 + (3.32 - 3.32i)T - 25iT^{2} \)
7 \( 1 + 4.04T + 49T^{2} \)
11 \( 1 + (6.82 + 6.82i)T + 121iT^{2} \)
13 \( 1 + (-4.29 - 4.29i)T + 169iT^{2} \)
17 \( 1 + 30.1T + 289T^{2} \)
19 \( 1 + (19.7 - 19.7i)T - 361iT^{2} \)
23 \( 1 - 28.2T + 529T^{2} \)
29 \( 1 + (-21.3 - 21.3i)T + 841iT^{2} \)
31 \( 1 + 38.0iT - 961T^{2} \)
37 \( 1 + (42.8 - 42.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 48.2iT - 1.68e3T^{2} \)
43 \( 1 + (-32.6 - 32.6i)T + 1.84e3iT^{2} \)
47 \( 1 + 15.8iT - 2.20e3T^{2} \)
53 \( 1 + (-0.476 + 0.476i)T - 2.80e3iT^{2} \)
59 \( 1 + (9.97 + 9.97i)T + 3.48e3iT^{2} \)
61 \( 1 + (-37.9 - 37.9i)T + 3.72e3iT^{2} \)
67 \( 1 + (-20.0 + 20.0i)T - 4.48e3iT^{2} \)
71 \( 1 + 40.0T + 5.04e3T^{2} \)
73 \( 1 + 30.8iT - 5.32e3T^{2} \)
79 \( 1 + 130. iT - 6.24e3T^{2} \)
83 \( 1 + (-2.26 + 2.26i)T - 6.88e3iT^{2} \)
89 \( 1 + 72.2iT - 7.92e3T^{2} \)
97 \( 1 + 112.T + 9.40e3T^{2} \)
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

−13.46594257750043330574705781575, −12.86454994420864002996349298724, −11.36614311110453062510149452872, −10.40119024899604175851984078611, −9.013933504717810751072804773983, −8.139335946670904217562553244442, −6.91572204326723053211699795222, −6.18397529816965881690424606921, −4.56435458000215151691236704003, −3.26014392114705490427152221697, 0.26512459208611921806128814013, 2.49071078637066401221928321429, 4.10837837550872020640314920001, 5.05135856991110003713158344368, 6.87193336410669702094728097777, 8.493592353112041771357344203437, 9.073006888365541010670787930834, 10.47272254368801208944807204469, 11.16450484398519243153680764185, 12.50368486429840415485058297547

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