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Properties

Label 2-24e2-144.131-c1-0-11
Degree $2$
Conductor $576$
Sign $0.552 - 0.833i$
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  + (0.800 + 1.53i)3-s + (3.73 − 1.00i)5-s + (1.68 + 2.91i)7-s + (−1.71 + 2.45i)9-s + (−0.211 − 0.0566i)11-s + (−2.71 + 0.727i)13-s + (4.52 + 4.93i)15-s − 4.23i·17-s + (1.12 − 1.12i)19-s + (−3.13 + 4.91i)21-s + (−3.33 − 1.92i)23-s + (8.61 − 4.97i)25-s + (−5.15 − 0.675i)27-s + (2.03 + 0.545i)29-s + (−7.21 − 4.16i)31-s + ⋯
L(s)  = 1  + (0.461 + 0.886i)3-s + (1.67 − 0.447i)5-s + (0.635 + 1.10i)7-s + (−0.573 + 0.819i)9-s + (−0.0637 − 0.0170i)11-s + (−0.753 + 0.201i)13-s + (1.16 + 1.27i)15-s − 1.02i·17-s + (0.257 − 0.257i)19-s + (−0.683 + 1.07i)21-s + (−0.695 − 0.401i)23-s + (1.72 − 0.994i)25-s + (−0.991 − 0.129i)27-s + (0.378 + 0.101i)29-s + (−1.29 − 0.747i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.552 - 0.833i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.552 - 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95551 + 1.05032i\)
\(L(\frac12)\) \(\approx\) \(1.95551 + 1.05032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.800 - 1.53i)T \)
good5 \( 1 + (-3.73 + 1.00i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.68 - 2.91i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.211 + 0.0566i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.71 - 0.727i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 4.23iT - 17T^{2} \)
19 \( 1 + (-1.12 + 1.12i)T - 19iT^{2} \)
23 \( 1 + (3.33 + 1.92i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.03 - 0.545i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (7.21 + 4.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.66 - 2.66i)T - 37iT^{2} \)
41 \( 1 + (-1.70 + 2.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.25 + 4.68i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.34 - 4.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.58 - 7.58i)T + 53iT^{2} \)
59 \( 1 + (-1.43 - 5.34i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.33 - 8.69i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.38 + 5.17i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.53iT - 71T^{2} \)
73 \( 1 + 3.22iT - 73T^{2} \)
79 \( 1 + (4.98 - 2.87i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.20 - 4.50i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 2.96T + 89T^{2} \)
97 \( 1 + (7.63 + 13.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.58437008563707037807992523713, −9.809050466032235568865202103572, −9.125939598664344413595810625887, −8.706439696321817551624071086999, −7.39825145926688700755914521657, −5.85642994258692817610333589129, −5.33898246894632159597463453727, −4.52786069975817181346640650466, −2.69902128539880044262103520326, −2.03457254899328879619115265358, 1.43298605261018188009248733301, 2.28337247768989350852916590186, 3.65580065690701510495747173769, 5.24572204901767467560308400187, 6.15176838702199831356823972587, 7.03461029236704376286589115614, 7.76190106402547114515571427625, 8.804794972834397283752937043724, 9.858773836976792087662631201508, 10.38899154044749681887574870645

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