↑

Properties

Label 2-12e2-16.11-c2-0-0
Degree $2$
Conductor $144$
Sign $-0.368 - 0.929i$
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  + (−1.10 − 1.66i)2-s + (−1.55 + 3.68i)4-s + (−4.23 − 4.23i)5-s − 0.262·7-s + (7.86 − 1.46i)8-s + (−2.38 + 11.7i)10-s + (−8.60 + 8.60i)11-s + (−15.9 + 15.9i)13-s + (0.289 + 0.437i)14-s + (−11.1 − 11.4i)16-s − 3.51·17-s + (10.7 + 10.7i)19-s + (22.2 − 9.00i)20-s + (23.8 + 4.84i)22-s − 16.4·23-s + ⋯
L(s)  = 1  + (−0.552 − 0.833i)2-s + (−0.389 + 0.920i)4-s + (−0.847 − 0.847i)5-s − 0.0374·7-s + (0.983 − 0.183i)8-s + (−0.238 + 1.17i)10-s + (−0.782 + 0.782i)11-s + (−1.23 + 1.23i)13-s + (0.0206 + 0.0312i)14-s + (−0.695 − 0.718i)16-s − 0.206·17-s + (0.566 + 0.566i)19-s + (1.11 − 0.450i)20-s + (1.08 + 0.220i)22-s − 0.717·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.368 - 0.929i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.368 - 0.929i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0521591 + 0.0767476i\)
\(L(\frac12)\) \(\approx\) \(0.0521591 + 0.0767476i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 + 1.66i)T \)
3 \( 1 \)
good5 \( 1 + (4.23 + 4.23i)T + 25iT^{2} \)
7 \( 1 + 0.262T + 49T^{2} \)
11 \( 1 + (8.60 - 8.60i)T - 121iT^{2} \)
13 \( 1 + (15.9 - 15.9i)T - 169iT^{2} \)
17 \( 1 + 3.51T + 289T^{2} \)
19 \( 1 + (-10.7 - 10.7i)T + 361iT^{2} \)
23 \( 1 + 16.4T + 529T^{2} \)
29 \( 1 + (25.9 - 25.9i)T - 841iT^{2} \)
31 \( 1 + 46.2iT - 961T^{2} \)
37 \( 1 + (2.99 + 2.99i)T + 1.36e3iT^{2} \)
41 \( 1 - 21.9iT - 1.68e3T^{2} \)
43 \( 1 + (-48.7 + 48.7i)T - 1.84e3iT^{2} \)
47 \( 1 + 70.7iT - 2.20e3T^{2} \)
53 \( 1 + (52.8 + 52.8i)T + 2.80e3iT^{2} \)
59 \( 1 + (61.7 - 61.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (22.9 - 22.9i)T - 3.72e3iT^{2} \)
67 \( 1 + (54.9 + 54.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 84.2T + 5.04e3T^{2} \)
73 \( 1 - 78.0iT - 5.32e3T^{2} \)
79 \( 1 - 59.2iT - 6.24e3T^{2} \)
83 \( 1 + (-111. - 111. i)T + 6.88e3iT^{2} \)
89 \( 1 + 34.5iT - 7.92e3T^{2} \)
97 \( 1 + 66.0T + 9.40e3T^{2} \)
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

−12.67580392398541784551266670055, −12.16680017648151594038813805706, −11.31472944728441299530592527375, −9.976353433986220045117449951954, −9.226084885310038655607540759667, −8.012149020822598077333426851001, −7.26908386705382203910499191921, −4.94763855256105552752526152276, −3.97880494035446277694560942556, −2.08555729701023357517556226214, 0.06934594399455137781923217643, 3.02577104298719131630271049721, 4.90487871202470549059411373433, 6.14856015073375830458198297130, 7.57223803162811548069388714518, 7.85192278824565803302701243452, 9.371039244010749094281954658986, 10.50012302563570141202945548183, 11.17708565861286630737760395413, 12.62096338589167132134811509900

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