↑

Properties

Label 2-12e2-16.3-c2-0-10
Degree $2$
Conductor $144$
Sign $0.992 - 0.118i$
Analytic cond. $3.92371$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.99 + 0.136i)2-s + (3.96 + 0.545i)4-s + (0.227 − 0.227i)5-s + 3.90·7-s + (7.83 + 1.63i)8-s + (0.485 − 0.423i)10-s + (−2.21 − 2.21i)11-s + (5.08 + 5.08i)13-s + (7.78 + 0.533i)14-s + (15.4 + 4.32i)16-s − 18.8·17-s + (11.7 − 11.7i)19-s + (1.02 − 0.777i)20-s + (−4.10 − 4.71i)22-s − 35.4·23-s + ⋯
L(s)  = 1  + (0.997 + 0.0683i)2-s + (0.990 + 0.136i)4-s + (0.0455 − 0.0455i)5-s + 0.557·7-s + (0.978 + 0.203i)8-s + (0.0485 − 0.0423i)10-s + (−0.200 − 0.200i)11-s + (0.391 + 0.391i)13-s + (0.556 + 0.0381i)14-s + (0.962 + 0.270i)16-s − 1.10·17-s + (0.619 − 0.619i)19-s + (0.0513 − 0.0388i)20-s + (−0.186 − 0.214i)22-s − 1.54·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.118i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.992 - 0.118i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.992 - 0.118i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.65538 + 0.158131i\)
\(L(\frac12)\) \(\approx\) \(2.65538 + 0.158131i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 - 0.136i)T \)
3 \( 1 \)
good5 \( 1 + (-0.227 + 0.227i)T - 25iT^{2} \)
7 \( 1 - 3.90T + 49T^{2} \)
11 \( 1 + (2.21 + 2.21i)T + 121iT^{2} \)
13 \( 1 + (-5.08 - 5.08i)T + 169iT^{2} \)
17 \( 1 + 18.8T + 289T^{2} \)
19 \( 1 + (-11.7 + 11.7i)T - 361iT^{2} \)
23 \( 1 + 35.4T + 529T^{2} \)
29 \( 1 + (21.2 + 21.2i)T + 841iT^{2} \)
31 \( 1 + 35.9iT - 961T^{2} \)
37 \( 1 + (34.4 - 34.4i)T - 1.36e3iT^{2} \)
41 \( 1 - 44.1iT - 1.68e3T^{2} \)
43 \( 1 + (28.3 + 28.3i)T + 1.84e3iT^{2} \)
47 \( 1 + 32.8iT - 2.20e3T^{2} \)
53 \( 1 + (-42.1 + 42.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (-66.9 - 66.9i)T + 3.48e3iT^{2} \)
61 \( 1 + (-17.3 - 17.3i)T + 3.72e3iT^{2} \)
67 \( 1 + (-39.6 + 39.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 63.0T + 5.04e3T^{2} \)
73 \( 1 - 75.4iT - 5.32e3T^{2} \)
79 \( 1 + 59.1iT - 6.24e3T^{2} \)
83 \( 1 + (-71.2 + 71.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 150. iT - 7.92e3T^{2} \)
97 \( 1 - 51.5T + 9.40e3T^{2} \)
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

−13.26248854808688037287436470869, −11.69654123990637188401173486323, −11.35595873335775809684637340235, −10.01450923542141487065454218126, −8.507204170010164362458697115075, −7.35507486104927136242839369959, −6.16631246059428246888914662077, −4.99650297934341143281648174676, −3.79162701046179643073474097540, −2.06093838098110441675590358771, 1.98903513416336192340287079316, 3.68390363961744272196866493995, 4.95778691809659494038795536357, 6.08385511117958451685844557236, 7.32252784092575630510093965132, 8.468106056858048455071623460654, 10.13560400361951720401633892640, 10.98665749727189800194626675841, 11.99997529662955636719663633152, 12.83884797744054231507123379212

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