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Properties

Label 2-24e2-16.11-c4-0-11
Degree $2$
Conductor $576$
Sign $-0.757 - 0.652i$
Analytic cond. $59.5410$
Root an. cond. $7.71628$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (29.2 + 29.2i)5-s − 59.6·7-s + (−18.0 + 18.0i)11-s + (50.7 − 50.7i)13-s + 223.·17-s + (−14.7 − 14.7i)19-s + 739.·23-s + 1.08e3i·25-s + (−938. + 938. i)29-s + 938. i·31-s + (−1.74e3 − 1.74e3i)35-s + (263. + 263. i)37-s − 248. i·41-s + (−1.03e3 + 1.03e3i)43-s + 2.01e3i·47-s + ⋯
L(s)  = 1  + (1.16 + 1.16i)5-s − 1.21·7-s + (−0.149 + 0.149i)11-s + (0.300 − 0.300i)13-s + 0.774·17-s + (−0.0408 − 0.0408i)19-s + 1.39·23-s + 1.72i·25-s + (−1.11 + 1.11i)29-s + 0.976i·31-s + (−1.42 − 1.42i)35-s + (0.192 + 0.192i)37-s − 0.148i·41-s + (−0.559 + 0.559i)43-s + 0.913i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.652i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.757 - 0.652i$
Analytic conductor: \(59.5410\)
Root analytic conductor: \(7.71628\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :2),\ -0.757 - 0.652i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.549274191\)
\(L(\frac12)\) \(\approx\) \(1.549274191\)
\(L(3)\) 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 \( 1 \)
good5 \( 1 + (-29.2 - 29.2i)T + 625iT^{2} \)
7 \( 1 + 59.6T + 2.40e3T^{2} \)
11 \( 1 + (18.0 - 18.0i)T - 1.46e4iT^{2} \)
13 \( 1 + (-50.7 + 50.7i)T - 2.85e4iT^{2} \)
17 \( 1 - 223.T + 8.35e4T^{2} \)
19 \( 1 + (14.7 + 14.7i)T + 1.30e5iT^{2} \)
23 \( 1 - 739.T + 2.79e5T^{2} \)
29 \( 1 + (938. - 938. i)T - 7.07e5iT^{2} \)
31 \( 1 - 938. iT - 9.23e5T^{2} \)
37 \( 1 + (-263. - 263. i)T + 1.87e6iT^{2} \)
41 \( 1 + 248. iT - 2.82e6T^{2} \)
43 \( 1 + (1.03e3 - 1.03e3i)T - 3.41e6iT^{2} \)
47 \( 1 - 2.01e3iT - 4.87e6T^{2} \)
53 \( 1 + (833. + 833. i)T + 7.89e6iT^{2} \)
59 \( 1 + (2.22e3 - 2.22e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (341. - 341. i)T - 1.38e7iT^{2} \)
67 \( 1 + (4.84e3 + 4.84e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 4.18e3T + 2.54e7T^{2} \)
73 \( 1 + 9.07e3iT - 2.83e7T^{2} \)
79 \( 1 + 735. iT - 3.89e7T^{2} \)
83 \( 1 + (-1.44e3 - 1.44e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 5.07e3iT - 6.27e7T^{2} \)
97 \( 1 + 2.52e3T + 8.85e7T^{2} \)
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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

−10.49816336525906023763815468511, −9.622248912985642397772318160371, −9.062820054935513877712192788573, −7.54093574290005304964573851112, −6.72430092807723829399872098472, −6.08902138857828232411408223990, −5.15457338882828978272799088813, −3.32376928713289649140108160415, −2.90286235926297827132578200047, −1.45375479703395306347312276718, 0.37600030622888093693369865368, 1.55061302482274666984270876596, 2.83839458903551488049002089794, 4.10914597210055794104245044603, 5.38367099045672587631587309306, 5.93261878471976937736253844845, 6.93061174305252122466265179000, 8.193548032092322064749722499744, 9.237070699787255345199751103054, 9.539910241400754368760880040171

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