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Properties

Label 2-24e2-48.35-c3-0-8
Degree $2$
Conductor $576$
Sign $0.0142 - 0.999i$
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  + (3.62 + 3.62i)5-s + 33.7·7-s + (−44.1 + 44.1i)11-s + (42.1 + 42.1i)13-s − 17.6i·17-s + (−99.2 + 99.2i)19-s + 83.8i·23-s − 98.7i·25-s + (89.7 − 89.7i)29-s − 46.2i·31-s + (122. + 122. i)35-s + (−7.47 + 7.47i)37-s − 299.·41-s + (56.3 + 56.3i)43-s − 280.·47-s + ⋯
L(s)  = 1  + (0.324 + 0.324i)5-s + 1.82·7-s + (−1.21 + 1.21i)11-s + (0.898 + 0.898i)13-s − 0.251i·17-s + (−1.19 + 1.19i)19-s + 0.759i·23-s − 0.789i·25-s + (0.574 − 0.574i)29-s − 0.267i·31-s + (0.590 + 0.590i)35-s + (−0.0332 + 0.0332i)37-s − 1.13·41-s + (0.199 + 0.199i)43-s − 0.869·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0142 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0142 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.0142 - 0.999i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ 0.0142 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.180653735\)
\(L(\frac12)\) \(\approx\) \(2.180653735\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.62 - 3.62i)T + 125iT^{2} \)
7 \( 1 - 33.7T + 343T^{2} \)
11 \( 1 + (44.1 - 44.1i)T - 1.33e3iT^{2} \)
13 \( 1 + (-42.1 - 42.1i)T + 2.19e3iT^{2} \)
17 \( 1 + 17.6iT - 4.91e3T^{2} \)
19 \( 1 + (99.2 - 99.2i)T - 6.85e3iT^{2} \)
23 \( 1 - 83.8iT - 1.21e4T^{2} \)
29 \( 1 + (-89.7 + 89.7i)T - 2.43e4iT^{2} \)
31 \( 1 + 46.2iT - 2.97e4T^{2} \)
37 \( 1 + (7.47 - 7.47i)T - 5.06e4iT^{2} \)
41 \( 1 + 299.T + 6.89e4T^{2} \)
43 \( 1 + (-56.3 - 56.3i)T + 7.95e4iT^{2} \)
47 \( 1 + 280.T + 1.03e5T^{2} \)
53 \( 1 + (8.15 + 8.15i)T + 1.48e5iT^{2} \)
59 \( 1 + (193. - 193. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-127. - 127. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-110. + 110. i)T - 3.00e5iT^{2} \)
71 \( 1 - 1.07e3iT - 3.57e5T^{2} \)
73 \( 1 - 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 + 161. iT - 4.93e5T^{2} \)
83 \( 1 + (64.4 + 64.4i)T + 5.71e5iT^{2} \)
89 \( 1 - 177.T + 7.04e5T^{2} \)
97 \( 1 - 559.T + 9.12e5T^{2} \)
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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

−10.51936532325503832894127440784, −9.881996356583730255836449258027, −8.497009265869616003875025351854, −8.025872838926310479824195682264, −7.04107093313795568059085292232, −5.89811508779558789230348953916, −4.86213434383781549523968540723, −4.12794106411041520707196268524, −2.29422781684927942489512399565, −1.59680969854467944416397697485, 0.63874686015072853132396754101, 1.89938770402641654427964712339, 3.20240140650052383546683533475, 4.74844683296314091327791700450, 5.27838767516651548731191722137, 6.31105251317681506517525257585, 7.72661264645287604827420694686, 8.442358230269454507443635852336, 8.800657915407311019560143303454, 10.63939292200164084617805211902

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