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Properties

Label 2-12e2-144.83-c1-0-16
Degree $2$
Conductor $144$
Sign $-0.429 + 0.902i$
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  + (−1.41 + 0.00754i)2-s + (−0.313 − 1.70i)3-s + (1.99 − 0.0213i)4-s + (0.752 − 2.80i)5-s + (0.455 + 2.40i)6-s + (−1.02 + 1.77i)7-s + (−2.82 + 0.0452i)8-s + (−2.80 + 1.06i)9-s + (−1.04 + 3.97i)10-s + (−0.447 − 1.67i)11-s + (−0.662 − 3.40i)12-s + (1.30 − 4.86i)13-s + (1.43 − 2.52i)14-s + (−5.02 − 0.402i)15-s + (3.99 − 0.0853i)16-s − 2.90i·17-s + ⋯
L(s)  = 1  + (−0.999 + 0.00533i)2-s + (−0.180 − 0.983i)3-s + (0.999 − 0.0106i)4-s + (0.336 − 1.25i)5-s + (0.186 + 0.982i)6-s + (−0.387 + 0.671i)7-s + (−0.999 + 0.0160i)8-s + (−0.934 + 0.355i)9-s + (−0.329 + 1.25i)10-s + (−0.135 − 0.503i)11-s + (−0.191 − 0.981i)12-s + (0.361 − 1.34i)13-s + (0.384 − 0.673i)14-s + (−1.29 − 0.103i)15-s + (0.999 − 0.0213i)16-s − 0.705i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.429 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.429 + 0.902i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.429 + 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.339982 - 0.538300i\)
\(L(\frac12)\) \(\approx\) \(0.339982 - 0.538300i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.00754i)T \)
3 \( 1 + (0.313 + 1.70i)T \)
good5 \( 1 + (-0.752 + 2.80i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.02 - 1.77i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.447 + 1.67i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.30 + 4.86i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + 2.90iT - 17T^{2} \)
19 \( 1 + (3.23 - 3.23i)T - 19iT^{2} \)
23 \( 1 + (0.831 - 0.480i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.48 - 5.55i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-8.00 + 4.62i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.18 + 8.18i)T - 37iT^{2} \)
41 \( 1 + (-4.06 - 7.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.38 - 0.907i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.60 + 2.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.65 - 1.65i)T + 53iT^{2} \)
59 \( 1 + (-8.27 - 2.21i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.06 + 0.284i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (9.36 + 2.50i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.40iT - 71T^{2} \)
73 \( 1 + 0.312iT - 73T^{2} \)
79 \( 1 + (-1.15 - 0.669i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-14.5 + 3.90i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 7.86T + 89T^{2} \)
97 \( 1 + (6.48 - 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.64836364723495185213952018042, −11.90956070372091313187269715250, −10.71423218425151285256770656710, −9.410928773118704156846279120470, −8.490785500893171219056380017923, −7.83759667611157254614431202749, −6.22848464915855775582003002158, −5.51265407999154563884287564791, −2.66551696015925927623311994784, −0.912648987097251171258329858713, 2.62031311066308787272523446033, 4.14674897920286938348567191603, 6.29211717740911430183253881545, 6.84827034922956150751494261240, 8.422945494547309375975244731977, 9.636888386761107643871103014948, 10.28424010572560400393537677784, 10.96925171983850410834580893286, 11.88465917754129213263855909308, 13.65764989342119935547727931962

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