Properties

Label 2-24e2-8.5-c3-0-20
Degree $2$
Conductor $576$
Sign $0.965 + 0.258i$
Analytic cond. $33.9851$
Root an. cond. $5.82967$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.8i·5-s + 27.7·7-s − 42i·11-s − 41.5i·13-s + 6·17-s − 94i·19-s + 138.·23-s − 66.9·25-s − 235. i·29-s − 110.·31-s + 383. i·35-s − 13.8i·37-s + 54·41-s + 442i·43-s + 55.4·47-s + ⋯
L(s)  = 1  + 1.23i·5-s + 1.49·7-s − 1.15i·11-s − 0.886i·13-s + 0.0856·17-s − 1.13i·19-s + 1.25·23-s − 0.535·25-s − 1.50i·29-s − 0.642·31-s + 1.85i·35-s − 0.0615i·37-s + 0.205·41-s + 1.56i·43-s + 0.172·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.965 + 0.258i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ 0.965 + 0.258i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.384477852\)
\(L(\frac12)\) \(\approx\) \(2.384477852\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5 \( 1 - 13.8iT - 125T^{2} \)
7 \( 1 - 27.7T + 343T^{2} \)
11 \( 1 + 42iT - 1.33e3T^{2} \)
13 \( 1 + 41.5iT - 2.19e3T^{2} \)
17 \( 1 - 6T + 4.91e3T^{2} \)
19 \( 1 + 94iT - 6.85e3T^{2} \)
23 \( 1 - 138.T + 1.21e4T^{2} \)
29 \( 1 + 235. iT - 2.43e4T^{2} \)
31 \( 1 + 110.T + 2.97e4T^{2} \)
37 \( 1 + 13.8iT - 5.06e4T^{2} \)
41 \( 1 - 54T + 6.89e4T^{2} \)
43 \( 1 - 442iT - 7.95e4T^{2} \)
47 \( 1 - 55.4T + 1.03e5T^{2} \)
53 \( 1 - 69.2iT - 1.48e5T^{2} \)
59 \( 1 + 138iT - 2.05e5T^{2} \)
61 \( 1 + 429. iT - 2.26e5T^{2} \)
67 \( 1 - 178iT - 3.00e5T^{2} \)
71 \( 1 - 859.T + 3.57e5T^{2} \)
73 \( 1 - 434T + 3.89e5T^{2} \)
79 \( 1 + 166.T + 4.93e5T^{2} \)
83 \( 1 - 270iT - 5.71e5T^{2} \)
89 \( 1 - 1.18e3T + 7.04e5T^{2} \)
97 \( 1 + 1.23e3T + 9.12e5T^{2} \)
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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

−10.65776379294551890998423571954, −9.412250915919088236279983993568, −8.316355790751114649733892518232, −7.70168313471833880083838158245, −6.71757344205006041229626385007, −5.68413717291761578564400274079, −4.72814698701715316749722775213, −3.33758843861140782091451793330, −2.44115526656448465811245005684, −0.822068487430762140239487582692, 1.23631755436381504838806195551, 1.95039949411512395902018752408, 3.96072169000761319483131538215, 4.87965823942515053499646583266, 5.33003076118195107657564546469, 6.93957732211319835574604653935, 7.76778563230328768032215914092, 8.709596281857208885047422869252, 9.217155211985771966507339693817, 10.39896381160457111872594052597

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