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Properties

Label 2-12e2-144.85-c1-0-5
Degree $2$
Conductor $144$
Sign $0.575 + 0.818i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.704 − 1.22i)2-s + (−1.58 + 0.705i)3-s + (−1.00 + 1.72i)4-s + (2.53 − 0.679i)5-s + (1.97 + 1.44i)6-s + (0.614 − 0.354i)7-s + (2.82 + 0.0162i)8-s + (2.00 − 2.23i)9-s + (−2.62 − 2.63i)10-s + (0.973 − 3.63i)11-s + (0.374 − 3.44i)12-s + (0.139 + 0.519i)13-s + (−0.867 − 0.503i)14-s + (−3.53 + 2.86i)15-s + (−1.97 − 3.47i)16-s + 6.08·17-s + ⋯
L(s)  = 1  + (−0.498 − 0.866i)2-s + (−0.913 + 0.407i)3-s + (−0.503 + 0.864i)4-s + (1.13 − 0.303i)5-s + (0.808 + 0.589i)6-s + (0.232 − 0.134i)7-s + (0.999 + 0.00574i)8-s + (0.668 − 0.743i)9-s + (−0.828 − 0.831i)10-s + (0.293 − 1.09i)11-s + (0.107 − 0.994i)12-s + (0.0385 + 0.143i)13-s + (−0.231 − 0.134i)14-s + (−0.912 + 0.739i)15-s + (−0.493 − 0.869i)16-s + 1.47·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.575 + 0.818i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.575 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.708164 - 0.367837i\)
\(L(\frac12)\) \(\approx\) \(0.708164 - 0.367837i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.704 + 1.22i)T \)
3 \( 1 + (1.58 - 0.705i)T \)
good5 \( 1 + (-2.53 + 0.679i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.614 + 0.354i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.973 + 3.63i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.139 - 0.519i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 6.08T + 17T^{2} \)
19 \( 1 + (1.86 - 1.86i)T - 19iT^{2} \)
23 \( 1 + (-4.94 - 2.85i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (9.68 + 2.59i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-2.14 + 3.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.75 - 3.75i)T + 37iT^{2} \)
41 \( 1 + (1.57 + 0.906i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.31 - 8.62i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.95 + 5.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.56 + 8.56i)T + 53iT^{2} \)
59 \( 1 + (5.19 - 1.39i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.655 - 0.175i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.96 + 7.33i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.51iT - 71T^{2} \)
73 \( 1 - 7.36iT - 73T^{2} \)
79 \( 1 + (-0.0143 - 0.0248i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-14.9 - 4.00i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 1.86iT - 89T^{2} \)
97 \( 1 + (-5.66 - 9.80i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.81886195855912268404585145178, −11.62884651971372287481035525133, −10.98337726526736320181009684489, −9.845054128117573662961398468993, −9.342511421515569724259545466240, −7.896963716613368415078528552417, −6.16215544202186391378074808249, −5.12147432719771685172587059820, −3.54271503787549617747719263396, −1.34832442943792713303613040546, 1.68909465017627971695645214205, 4.87878491827174999671862520135, 5.77830992702100462641140803255, 6.76170816881676651670885746277, 7.65124669019235691580927776968, 9.233239524626731375685100184877, 10.10957617972375670278827184436, 10.93116257913009834228424781045, 12.40666763365067039817778630736, 13.30222130722810664580293372993

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