Properties

Label 2-12e2-48.29-c2-0-15
Degree $2$
Conductor $144$
Sign $-0.878 + 0.477i$
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  + (1.13 − 1.64i)2-s + (−1.40 − 3.74i)4-s + (−3.90 − 3.90i)5-s − 0.778i·7-s + (−7.76 − 1.94i)8-s + (−10.8 + 1.97i)10-s + (−4.29 − 4.29i)11-s + (6.44 + 6.44i)13-s + (−1.28 − 0.886i)14-s + (−12.0 + 10.5i)16-s − 31.3i·17-s + (1.11 + 1.11i)19-s + (−9.11 + 20.1i)20-s + (−11.9 + 2.17i)22-s + 34.2·23-s + ⋯
L(s)  = 1  + (0.569 − 0.822i)2-s + (−0.352 − 0.935i)4-s + (−0.780 − 0.780i)5-s − 0.111i·7-s + (−0.970 − 0.242i)8-s + (−1.08 + 0.197i)10-s + (−0.390 − 0.390i)11-s + (0.495 + 0.495i)13-s + (−0.0914 − 0.0633i)14-s + (−0.751 + 0.659i)16-s − 1.84i·17-s + (0.0588 + 0.0588i)19-s + (−0.455 + 1.00i)20-s + (−0.542 + 0.0987i)22-s + 1.48·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.878 + 0.477i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.878 + 0.477i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.344589 - 1.35698i\)
\(L(\frac12)\) \(\approx\) \(0.344589 - 1.35698i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 + 1.64i)T \)
3 \( 1 \)
good5 \( 1 + (3.90 + 3.90i)T + 25iT^{2} \)
7 \( 1 + 0.778iT - 49T^{2} \)
11 \( 1 + (4.29 + 4.29i)T + 121iT^{2} \)
13 \( 1 + (-6.44 - 6.44i)T + 169iT^{2} \)
17 \( 1 + 31.3iT - 289T^{2} \)
19 \( 1 + (-1.11 - 1.11i)T + 361iT^{2} \)
23 \( 1 - 34.2T + 529T^{2} \)
29 \( 1 + (8.77 - 8.77i)T - 841iT^{2} \)
31 \( 1 - 50.8T + 961T^{2} \)
37 \( 1 + (29.3 - 29.3i)T - 1.36e3iT^{2} \)
41 \( 1 + 31.4T + 1.68e3T^{2} \)
43 \( 1 + (-55.9 + 55.9i)T - 1.84e3iT^{2} \)
47 \( 1 - 26.5iT - 2.20e3T^{2} \)
53 \( 1 + (9.76 + 9.76i)T + 2.80e3iT^{2} \)
59 \( 1 + (-54.3 - 54.3i)T + 3.48e3iT^{2} \)
61 \( 1 + (47.1 + 47.1i)T + 3.72e3iT^{2} \)
67 \( 1 + (66.1 + 66.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 75.9T + 5.04e3T^{2} \)
73 \( 1 - 24.1iT - 5.32e3T^{2} \)
79 \( 1 + 80.5T + 6.24e3T^{2} \)
83 \( 1 + (82.6 - 82.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 82.6T + 7.92e3T^{2} \)
97 \( 1 - 48.9T + 9.40e3T^{2} \)
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

−12.26204607730651838398241947013, −11.63336214928781092011929844593, −10.69537440187877282076379398999, −9.372420209135913365631770871733, −8.504226462905895421686319594918, −6.95746411391035475315790329198, −5.30753596143891997267536745470, −4.39574294606868086947042368862, −2.97802941749796293028673485251, −0.805776275585091904266646882607, 3.06766641883539091135159865853, 4.23225222928364530022099902415, 5.72221439313811392021681089026, 6.85844251816881854252633104478, 7.81923761993847393075177109553, 8.724733368185591049104641484318, 10.39351504331216604836529121940, 11.38343938405147538683875880359, 12.53946985147244361262140988679, 13.27616340486174363427505890658

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