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Properties

Label 2-24e2-48.35-c3-0-16
Degree $2$
Conductor $576$
Sign $-0.0369 + 0.999i$
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  + (6.31 + 6.31i)5-s − 16.2·7-s + (−48.1 + 48.1i)11-s + (−8.61 − 8.61i)13-s − 53.2i·17-s + (55.5 − 55.5i)19-s − 66.9i·23-s − 45.2i·25-s + (126. − 126. i)29-s − 121. i·31-s + (−102. − 102. i)35-s + (−250. + 250. i)37-s + 402.·41-s + (−187. − 187. i)43-s + 96.1·47-s + ⋯
L(s)  = 1  + (0.564 + 0.564i)5-s − 0.878·7-s + (−1.31 + 1.31i)11-s + (−0.183 − 0.183i)13-s − 0.759i·17-s + (0.670 − 0.670i)19-s − 0.607i·23-s − 0.361i·25-s + (0.809 − 0.809i)29-s − 0.701i·31-s + (−0.496 − 0.496i)35-s + (−1.11 + 1.11i)37-s + 1.53·41-s + (−0.664 − 0.664i)43-s + 0.298·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0369 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0369 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.0369 + 0.999i$
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: \((2,\ 576,\ (\ :3/2),\ -0.0369 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9271845296\)
\(L(\frac12)\) \(\approx\) \(0.9271845296\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.31 - 6.31i)T + 125iT^{2} \)
7 \( 1 + 16.2T + 343T^{2} \)
11 \( 1 + (48.1 - 48.1i)T - 1.33e3iT^{2} \)
13 \( 1 + (8.61 + 8.61i)T + 2.19e3iT^{2} \)
17 \( 1 + 53.2iT - 4.91e3T^{2} \)
19 \( 1 + (-55.5 + 55.5i)T - 6.85e3iT^{2} \)
23 \( 1 + 66.9iT - 1.21e4T^{2} \)
29 \( 1 + (-126. + 126. i)T - 2.43e4iT^{2} \)
31 \( 1 + 121. iT - 2.97e4T^{2} \)
37 \( 1 + (250. - 250. i)T - 5.06e4iT^{2} \)
41 \( 1 - 402.T + 6.89e4T^{2} \)
43 \( 1 + (187. + 187. i)T + 7.95e4iT^{2} \)
47 \( 1 - 96.1T + 1.03e5T^{2} \)
53 \( 1 + (90.3 + 90.3i)T + 1.48e5iT^{2} \)
59 \( 1 + (-488. + 488. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-378. - 378. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-223. + 223. i)T - 3.00e5iT^{2} \)
71 \( 1 + 231. iT - 3.57e5T^{2} \)
73 \( 1 + 265. iT - 3.89e5T^{2} \)
79 \( 1 + 604. iT - 4.93e5T^{2} \)
83 \( 1 + (351. + 351. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + 1.85e3T + 9.12e5T^{2} \)
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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

−9.896689483162708183340199684429, −9.671151164710052469923244510254, −8.255369551663511039965349465892, −7.22481210455064864659127861250, −6.61356171940925254850517924024, −5.46373773275253237317109528218, −4.54077718306378700480927431872, −2.94055842927925064716235355996, −2.32079416961841631053753840821, −0.28430392802045467571895258911, 1.20095028943305734283972390012, 2.75627048521832398425734833799, 3.71130825958725806639413406982, 5.32852439980227623890245707929, 5.72257382041963036081455692924, 6.88879355463175223982043964999, 8.030613080815416922585420561432, 8.803179120832491939561563315764, 9.697748166420303951211887729336, 10.44644580539139763662597444414

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