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Properties

Label 2-24e2-144.85-c1-0-8
Degree $2$
Conductor $576$
Sign $0.737 + 0.675i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
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  − 1.73·3-s + (−1.86 + 0.5i)5-s + (−3.86 + 2.23i)7-s + 2.99·9-s + (0.5 − 1.86i)11-s + (−0.598 − 2.23i)13-s + (3.23 − 0.866i)15-s + 4·17-s + (3 − 3i)19-s + (6.69 − 3.86i)21-s + (5.59 + 3.23i)23-s + (−1.09 + 0.633i)25-s − 5.19·27-s + (−0.866 − 0.232i)29-s + (4.59 − 7.96i)31-s + ⋯
L(s)  = 1  − 1.00·3-s + (−0.834 + 0.223i)5-s + (−1.46 + 0.843i)7-s + 0.999·9-s + (0.150 − 0.562i)11-s + (−0.165 − 0.619i)13-s + (0.834 − 0.223i)15-s + 0.970·17-s + (0.688 − 0.688i)19-s + (1.46 − 0.843i)21-s + (1.16 + 0.673i)23-s + (−0.219 + 0.126i)25-s − 1.00·27-s + (−0.160 − 0.0430i)29-s + (0.825 − 1.43i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.737 + 0.675i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.737 + 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.584245 - 0.227200i\)
\(L(\frac12)\) \(\approx\) \(0.584245 - 0.227200i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T \)
good5 \( 1 + (1.86 - 0.5i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (3.86 - 2.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 1.86i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.598 + 2.23i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (-3 + 3i)T - 19iT^{2} \)
23 \( 1 + (-5.59 - 3.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.866 + 0.232i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-4.59 + 7.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.26 + 4.26i)T + 37iT^{2} \)
41 \( 1 + (-0.696 - 0.401i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.69 + 6.33i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.598 - 1.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.73 - 5.73i)T + 53iT^{2} \)
59 \( 1 + (-1.5 + 0.401i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.13 + 0.571i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.23 - 8.33i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 + 7.46iT - 73T^{2} \)
79 \( 1 + (-0.866 - 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-14.1 - 3.79i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 15.8iT - 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.72298775923995625158637915099, −9.779023370194734272109083621970, −9.086897391397032922471223686478, −7.71037561023842606939950159584, −6.97870671830929623457396781753, −5.92065269526953227821907328908, −5.34878075618679775552245540164, −3.81038732303082324747259825074, −2.93578004228517419672645321092, −0.54060939883141623164365768143, 0.997629187816639011412711282623, 3.32459802285892343151666224042, 4.20893674299681912956562086341, 5.21670761898454019895349875639, 6.55039978667316210542565651752, 6.98380485137475481012919538917, 7.941622420644210467159155748150, 9.380945225577482841218384477712, 10.05834035081467613523584706962, 10.73229265721196713980706161290

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