Properties

Label 2-12e2-36.23-c1-0-2
Degree $2$
Conductor $144$
Sign $0.984 - 0.173i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·3-s + (1.5 + 0.866i)5-s + (−2.59 + 1.5i)7-s + 2.99·9-s + (−2.59 − 4.5i)11-s + (−0.5 + 0.866i)13-s + (2.59 + 1.49i)15-s − 3.46i·17-s + 6i·19-s + (−4.5 + 2.59i)21-s + (2.59 − 4.5i)23-s + (−1 − 1.73i)25-s + 5.19·27-s + (−7.5 + 4.33i)29-s + (−2.59 − 1.5i)31-s + ⋯
L(s)  = 1  + 1.00·3-s + (0.670 + 0.387i)5-s + (−0.981 + 0.566i)7-s + 0.999·9-s + (−0.783 − 1.35i)11-s + (−0.138 + 0.240i)13-s + (0.670 + 0.387i)15-s − 0.840i·17-s + 1.37i·19-s + (−0.981 + 0.566i)21-s + (0.541 − 0.938i)23-s + (−0.200 − 0.346i)25-s + 1.00·27-s + (−1.39 + 0.804i)29-s + (−0.466 − 0.269i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44601 + 0.126509i\)
\(L(\frac12)\) \(\approx\) \(1.44601 + 0.126509i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
good5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.59 - 1.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.59 + 4.5i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + (-2.59 + 4.5i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.5 - 4.33i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.59 + 1.5i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.59 - 1.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.59 - 4.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + (2.59 - 4.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.79 - 4.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (-12.9 + 7.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.59 - 4.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.46iT - 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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

−13.25644918695345062455243655222, −12.45553668997881141967631147022, −10.88113874778016636316576246470, −9.861275374708956164120571079545, −9.076031765902543733084110168792, −8.008019579604134389172515308716, −6.65780360780015032693371357846, −5.54237733388065670401573669277, −3.48429164753583361322936142108, −2.46632656362164108024302273549, 2.13731651477474007062868200648, 3.69268206051162075448026450011, 5.17852515331783687498942647739, 6.87550645366045446450862709724, 7.71201406813773082747995185529, 9.247501654964926554511061083515, 9.694107273038700793747814262090, 10.72482071138261257623531497156, 12.64312149583815458770661577064, 13.09496732652307132032936546049

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