Properties

Label 2-12e2-144.13-c1-0-8
Degree $2$
Conductor $144$
Sign $0.978 - 0.206i$
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  + (1.24 − 0.677i)2-s + (−1.37 + 1.05i)3-s + (1.08 − 1.68i)4-s + (1.10 + 4.10i)5-s + (−0.993 + 2.23i)6-s + (1.63 − 0.942i)7-s + (0.201 − 2.82i)8-s + (0.782 − 2.89i)9-s + (4.15 + 4.35i)10-s + (−1.18 − 0.317i)11-s + (0.284 + 3.45i)12-s + (−2.31 + 0.620i)13-s + (1.38 − 2.27i)14-s + (−5.84 − 4.49i)15-s + (−1.66 − 3.63i)16-s + 3.44·17-s + ⋯
L(s)  = 1  + (0.877 − 0.479i)2-s + (−0.794 + 0.607i)3-s + (0.540 − 0.841i)4-s + (0.492 + 1.83i)5-s + (−0.405 + 0.914i)6-s + (0.617 − 0.356i)7-s + (0.0713 − 0.997i)8-s + (0.260 − 0.965i)9-s + (1.31 + 1.37i)10-s + (−0.356 − 0.0956i)11-s + (0.0820 + 0.996i)12-s + (−0.641 + 0.171i)13-s + (0.370 − 0.608i)14-s + (−1.50 − 1.15i)15-s + (−0.415 − 0.909i)16-s + 0.835·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.978 - 0.206i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.978 - 0.206i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50877 + 0.157494i\)
\(L(\frac12)\) \(\approx\) \(1.50877 + 0.157494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 + 0.677i)T \)
3 \( 1 + (1.37 - 1.05i)T \)
good5 \( 1 + (-1.10 - 4.10i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.63 + 0.942i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.18 + 0.317i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.31 - 0.620i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 + (4.17 + 4.17i)T + 19iT^{2} \)
23 \( 1 + (1.34 + 0.778i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.343 - 1.28i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-1.25 + 2.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.77 + 5.77i)T - 37iT^{2} \)
41 \( 1 + (3.43 + 1.98i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.96 + 0.793i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.230 - 0.399i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.61 + 5.61i)T - 53iT^{2} \)
59 \( 1 + (-2.92 - 10.9i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.27 + 4.74i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.286 + 0.0767i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + 0.0279iT - 73T^{2} \)
79 \( 1 + (-2.19 - 3.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.915 - 3.41i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 4.74iT - 89T^{2} \)
97 \( 1 + (-4.17 - 7.22i)T + (-48.5 + 84.0i)T^{2} \)
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

−13.17376512029052294466252367629, −11.83131592373927448951581628524, −11.04682325291565884051669124484, −10.47136883640045362577237468279, −9.722945488846536939504144544741, −7.28004840454809747609471927865, −6.39643436386473494537511845035, −5.32308746499460993410728493269, −3.95980682117106618698672210865, −2.53893211741526112275953983295, 1.83887080298154488061909788628, 4.58899903274983418734625410405, 5.28656566790615517194370453393, 6.12186633919418477995397928828, 7.80367537517233734968411403751, 8.420505800009734045446848042998, 10.08694523815291026411273987996, 11.69339906915879771275894804140, 12.30826344830108950176337059350, 12.92086278500180252033575180902

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