↑

Properties

Label 2-12e2-9.5-c2-0-1
Degree $2$
Conductor $144$
Sign $-0.418 - 0.908i$
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  + (−1.83 + 2.37i)3-s + (3.44 + 1.98i)5-s + (1.80 + 3.12i)7-s + (−2.29 − 8.70i)9-s + (−11.8 + 6.83i)11-s + (−8.96 + 15.5i)13-s + (−11.0 + 4.54i)15-s + 21.6i·17-s + 23.4·19-s + (−10.7 − 1.43i)21-s + (−13.0 − 7.54i)23-s + (−4.59 − 7.96i)25-s + (24.8 + 10.4i)27-s + (−20.4 + 11.8i)29-s + (23.5 − 40.7i)31-s + ⋯
L(s)  = 1  + (−0.610 + 0.792i)3-s + (0.688 + 0.397i)5-s + (0.257 + 0.446i)7-s + (−0.254 − 0.966i)9-s + (−1.07 + 0.621i)11-s + (−0.689 + 1.19i)13-s + (−0.735 + 0.302i)15-s + 1.27i·17-s + 1.23·19-s + (−0.511 − 0.0683i)21-s + (−0.568 − 0.327i)23-s + (−0.183 − 0.318i)25-s + (0.921 + 0.388i)27-s + (−0.706 + 0.407i)29-s + (0.758 − 1.31i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.418 - 0.908i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.418 - 0.908i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.595151 + 0.929981i\)
\(L(\frac12)\) \(\approx\) \(0.595151 + 0.929981i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.83 - 2.37i)T \)
good5 \( 1 + (-3.44 - 1.98i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-1.80 - 3.12i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (11.8 - 6.83i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (8.96 - 15.5i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 21.6iT - 289T^{2} \)
19 \( 1 - 23.4T + 361T^{2} \)
23 \( 1 + (13.0 + 7.54i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (20.4 - 11.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-23.5 + 40.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 54.6T + 1.36e3T^{2} \)
41 \( 1 + (-24.6 - 14.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-23.8 - 41.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (30.5 - 17.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 65.0iT - 2.80e3T^{2} \)
59 \( 1 + (-76.0 - 43.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (6.46 + 11.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (1.55 - 2.69i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 49.5iT - 5.04e3T^{2} \)
73 \( 1 - 102.T + 5.32e3T^{2} \)
79 \( 1 + (-18.7 - 32.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (14.6 - 8.48i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 14.1iT - 7.92e3T^{2} \)
97 \( 1 + (24.1 + 41.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−13.10193173359398114796535020572, −12.00351378252285498018909094674, −11.11793052815635269069042602691, −9.973534130347541036973321975303, −9.521090795573404386640242810244, −7.928900335629427390720649447179, −6.43173804428870623595764746872, −5.45470411560410169306692996366, −4.28155644086081703617838582887, −2.35369857648489357740883364065, 0.78159255934550474528862277978, 2.68660489964277520565897877113, 5.13931000974952475288315839515, 5.67642892470472904451254285043, 7.32344784942457284293068375015, 7.982181173563890547337688817051, 9.592918553425509019044770326632, 10.58746545627840407390323952243, 11.59994387913210285103841351796, 12.64532798643882399327528949027

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