↑

Properties

Label 2-12e2-36.11-c1-0-1
Degree $2$
Conductor $144$
Sign $0.642 - 0.766i$
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  + 1.73i·3-s + (1.5 − 0.866i)5-s + (1.5 + 0.866i)7-s − 2.99·9-s + (−1.5 + 2.59i)11-s + (2.5 + 4.33i)13-s + (1.49 + 2.59i)15-s − 6.92i·17-s − 3.46i·19-s + (−1.49 + 2.59i)21-s + (−4.5 − 7.79i)23-s + (−1 + 1.73i)25-s − 5.19i·27-s + (1.5 + 0.866i)29-s + (4.5 − 2.59i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (0.670 − 0.387i)5-s + (0.566 + 0.327i)7-s − 0.999·9-s + (−0.452 + 0.783i)11-s + (0.693 + 1.20i)13-s + (0.387 + 0.670i)15-s − 1.68i·17-s − 0.794i·19-s + (−0.327 + 0.566i)21-s + (−0.938 − 1.62i)23-s + (−0.200 + 0.346i)25-s − 0.999i·27-s + (0.278 + 0.160i)29-s + (0.808 − 0.466i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09845 + 0.512216i\)
\(L(\frac12)\) \(\approx\) \(1.09845 + 0.512216i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
good5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.92iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.5 + 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.5 + 2.59i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.5 - 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-7.5 - 4.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.5 - 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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

−13.49078372668303855757759986490, −11.99739986298116728067896772607, −11.23134466751041772860622014141, −9.991315260040037650100089460191, −9.269350462714940698997241122082, −8.323173913648606180816209590042, −6.62300201738300098617533839487, −5.19156925697800759713470792836, −4.43707803328412669519358222968, −2.41336522341264974174290916616, 1.63805317734805943639502836416, 3.38446480808265858762667512549, 5.65254390586573754776229900189, 6.25801030061991551753669093673, 7.909440859469040367913132712076, 8.308151510695133744638530031863, 10.14460845756410930506594181964, 10.90212705694862630101407144930, 12.04276712794267890174764583776, 13.19624302813122707111145947699

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