Properties

Label 2-12e2-9.4-c1-0-1
Degree $2$
Conductor $144$
Sign $-0.173 - 0.984i$
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  + 1.73i·3-s + (−1.5 + 2.59i)5-s + (−0.5 − 0.866i)7-s − 2.99·9-s + (1.5 + 2.59i)11-s + (0.5 − 0.866i)13-s + (−4.5 − 2.59i)15-s + 6·17-s + 4·19-s + (1.49 − 0.866i)21-s + (−1.5 + 2.59i)23-s + (−2 − 3.46i)25-s − 5.19i·27-s + (−1.5 − 2.59i)29-s + (2.5 − 4.33i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (−0.670 + 1.16i)5-s + (−0.188 − 0.327i)7-s − 0.999·9-s + (0.452 + 0.783i)11-s + (0.138 − 0.240i)13-s + (−1.16 − 0.670i)15-s + 1.45·17-s + 0.917·19-s + (0.327 − 0.188i)21-s + (−0.312 + 0.541i)23-s + (−0.400 − 0.692i)25-s − 0.999i·27-s + (−0.278 − 0.482i)29-s + (0.449 − 0.777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620791 + 0.739830i\)
\(L(\frac12)\) \(\approx\) \(0.620791 + 0.739830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 1.73iT \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (5.5 + 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.63939936948839066671965589658, −11.98128141849072251647111542140, −11.35052065175557841697075784163, −10.17307402423722297079296465044, −9.686296243206800738249221035342, −8.037768318414551450244332052876, −7.06442974047196561827728526342, −5.62145106335824226285795652209, −4.04531457728349794313523609829, −3.12456873246481322824104634200, 1.10309822942172362190392132849, 3.34725078321806087360246920646, 5.13111030909990043887555761680, 6.27944521460285198715481266591, 7.71215399010766595203941265039, 8.451398717742178608733058238491, 9.419095640760460642674684366510, 11.17521245136674454721478943905, 12.17938940567116157236683203657, 12.49753018174528950969373641326

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