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Properties

Label 2-24e2-36.7-c2-0-13
Degree $2$
Conductor $576$
Sign $0.987 + 0.160i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.89 + 0.795i)3-s + (−0.355 − 0.615i)5-s + (2.70 + 1.56i)7-s + (7.73 − 4.60i)9-s + (−14.3 − 8.30i)11-s + (9.17 + 15.8i)13-s + (1.51 + 1.49i)15-s − 9.69·17-s − 8.20i·19-s + (−9.06 − 2.36i)21-s + (1.94 − 1.12i)23-s + (12.2 − 21.2i)25-s + (−18.7 + 19.4i)27-s + (20.8 − 36.0i)29-s + (−21.6 + 12.4i)31-s + ⋯
L(s)  = 1  + (−0.964 + 0.265i)3-s + (−0.0710 − 0.123i)5-s + (0.386 + 0.223i)7-s + (0.859 − 0.511i)9-s + (−1.30 − 0.754i)11-s + (0.705 + 1.22i)13-s + (0.101 + 0.0998i)15-s − 0.570·17-s − 0.431i·19-s + (−0.431 − 0.112i)21-s + (0.0847 − 0.0489i)23-s + (0.489 − 0.848i)25-s + (−0.692 + 0.721i)27-s + (0.717 − 1.24i)29-s + (−0.697 + 0.402i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.987 + 0.160i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.987 + 0.160i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.167179593\)
\(L(\frac12)\) \(\approx\) \(1.167179593\)
\(L(2)\) 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 \( 1 + (2.89 - 0.795i)T \)
good5 \( 1 + (0.355 + 0.615i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-2.70 - 1.56i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (14.3 + 8.30i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-9.17 - 15.8i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 9.69T + 289T^{2} \)
19 \( 1 + 8.20iT - 361T^{2} \)
23 \( 1 + (-1.94 + 1.12i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-20.8 + 36.0i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (21.6 - 12.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 40.3T + 1.36e3T^{2} \)
41 \( 1 + (-25.6 - 44.5i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-56.6 - 32.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-29.2 - 16.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 90.6T + 2.80e3T^{2} \)
59 \( 1 + (-66.2 + 38.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (1.35 - 2.35i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-34.5 + 19.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 102. iT - 5.04e3T^{2} \)
73 \( 1 - 38.1T + 5.32e3T^{2} \)
79 \( 1 + (-94.4 - 54.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (113. + 65.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 38.0T + 7.92e3T^{2} \)
97 \( 1 + (12.1 - 21.1i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.81654087924467820374533665949, −9.709357922680699539578116746237, −8.766403210345558621238812512203, −7.86668181190866156083359275585, −6.64571945070352729922136439564, −5.91560715456244566475820897204, −4.89066166515532263133420081305, −4.10709722916793157082006918662, −2.45194575907924953574273896882, −0.72226661195392696119481537695, 0.890892422912187226533729607025, 2.43606049439942400061558970910, 4.02992766972515496114690746272, 5.21108521205783781921209527291, 5.73485135442226391156130017109, 7.10086678826113785964657925522, 7.61029326225972205615732844238, 8.658426105840680093241306753861, 10.02487965935024695825022403098, 10.74425597016962521893753827887

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