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Properties

Label 2-24e2-144.13-c1-0-13
Degree $2$
Conductor $576$
Sign $0.999 - 0.0436i$
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 + (−0.133 − 0.5i)5-s + (−2.13 + 1.23i)7-s + 2.99·9-s + (0.5 + 0.133i)11-s + (4.59 − 1.23i)13-s + (−0.232 − 0.866i)15-s + 4·17-s + (3 + 3i)19-s + (−3.69 + 2.13i)21-s + (0.401 + 0.232i)23-s + (4.09 − 2.36i)25-s + 5.19·27-s + (0.866 − 3.23i)29-s + (−0.598 + 1.03i)31-s + ⋯
L(s)  = 1  + 1.00·3-s + (−0.0599 − 0.223i)5-s + (−0.806 + 0.465i)7-s + 0.999·9-s + (0.150 + 0.0403i)11-s + (1.27 − 0.341i)13-s + (−0.0599 − 0.223i)15-s + 0.970·17-s + (0.688 + 0.688i)19-s + (−0.806 + 0.465i)21-s + (0.0838 + 0.0483i)23-s + (0.819 − 0.473i)25-s + 1.00·27-s + (0.160 − 0.600i)29-s + (−0.107 + 0.186i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.999 - 0.0436i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.999 - 0.0436i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05108 + 0.0447548i\)
\(L(\frac12)\) \(\approx\) \(2.05108 + 0.0447548i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.133 + 0.5i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (2.13 - 1.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.133i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-4.59 + 1.23i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (-3 - 3i)T + 19iT^{2} \)
23 \( 1 + (-0.401 - 0.232i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.866 + 3.23i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (0.598 - 1.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.73 - 7.73i)T - 37iT^{2} \)
41 \( 1 + (9.69 + 5.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.69 + 2.33i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.59 + 7.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.26 + 2.26i)T - 53iT^{2} \)
59 \( 1 + (-1.5 - 5.59i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.86 - 14.4i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.23 - 0.330i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 - 0.535iT - 73T^{2} \)
79 \( 1 + (0.866 + 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.16 - 11.7i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 11.8iT - 89T^{2} \)
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

−10.30911090600662174194478364653, −9.930529250948381807008971046346, −8.707924751300379573792867017802, −8.427324653128115175169979399667, −7.22278415579048469252497398337, −6.26951188858987718629264523414, −5.14998763732529670609669116467, −3.65851573183850815798317561239, −3.10205552391469698429463490078, −1.46631370867637442707384698035, 1.41062282073991015673508038785, 3.19185835578461986637065365405, 3.58397779739987362553225444999, 5.01640262029483827410255593502, 6.49352455696676240798388646126, 7.10152162371661289430432572657, 8.147120320661908595282009329820, 8.997684682875285590859830980602, 9.734027336407123501804329384146, 10.55580316793066198270580685775

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