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Properties

Label 2-24e2-48.35-c3-0-10
Degree $2$
Conductor $576$
Sign $0.231 - 0.972i$
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  + (9.40 + 9.40i)5-s + 3.57·7-s + (−3.36 + 3.36i)11-s + (26.9 + 26.9i)13-s + 12.7i·17-s + (50.0 − 50.0i)19-s + 208. i·23-s + 51.7i·25-s + (134. − 134. i)29-s − 80.1i·31-s + (33.5 + 33.5i)35-s + (−308. + 308. i)37-s − 172.·41-s + (87.0 + 87.0i)43-s + 525.·47-s + ⋯
L(s)  = 1  + (0.840 + 0.840i)5-s + 0.192·7-s + (−0.0921 + 0.0921i)11-s + (0.574 + 0.574i)13-s + 0.182i·17-s + (0.604 − 0.604i)19-s + 1.88i·23-s + 0.413i·25-s + (0.859 − 0.859i)29-s − 0.464i·31-s + (0.162 + 0.162i)35-s + (−1.37 + 1.37i)37-s − 0.658·41-s + (0.308 + 0.308i)43-s + 1.63·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.231 - 0.972i$
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: \((2,\ 576,\ (\ :3/2),\ 0.231 - 0.972i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.312897621\)
\(L(\frac12)\) \(\approx\) \(2.312897621\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.40 - 9.40i)T + 125iT^{2} \)
7 \( 1 - 3.57T + 343T^{2} \)
11 \( 1 + (3.36 - 3.36i)T - 1.33e3iT^{2} \)
13 \( 1 + (-26.9 - 26.9i)T + 2.19e3iT^{2} \)
17 \( 1 - 12.7iT - 4.91e3T^{2} \)
19 \( 1 + (-50.0 + 50.0i)T - 6.85e3iT^{2} \)
23 \( 1 - 208. iT - 1.21e4T^{2} \)
29 \( 1 + (-134. + 134. i)T - 2.43e4iT^{2} \)
31 \( 1 + 80.1iT - 2.97e4T^{2} \)
37 \( 1 + (308. - 308. i)T - 5.06e4iT^{2} \)
41 \( 1 + 172.T + 6.89e4T^{2} \)
43 \( 1 + (-87.0 - 87.0i)T + 7.95e4iT^{2} \)
47 \( 1 - 525.T + 1.03e5T^{2} \)
53 \( 1 + (-127. - 127. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-172. + 172. i)T - 2.05e5iT^{2} \)
61 \( 1 + (332. + 332. i)T + 2.26e5iT^{2} \)
67 \( 1 + (556. - 556. i)T - 3.00e5iT^{2} \)
71 \( 1 - 450. iT - 3.57e5T^{2} \)
73 \( 1 - 797. iT - 3.89e5T^{2} \)
79 \( 1 - 70.1iT - 4.93e5T^{2} \)
83 \( 1 + (-636. - 636. i)T + 5.71e5iT^{2} \)
89 \( 1 - 925.T + 7.04e5T^{2} \)
97 \( 1 - 1.26e3T + 9.12e5T^{2} \)
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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

−10.40233647337075852816481524035, −9.746519658882071625071594676928, −8.878720209675347087245277222902, −7.74565265369611473692148526042, −6.81060095662626559104618786358, −6.03898660044033058393234474135, −5.04858339218512518585962134222, −3.69368898786911581734460669956, −2.56765154500558605347457146298, −1.39338071168772411471190611002, 0.72387029290057977352718793634, 1.89214369620304398900072535862, 3.28495353057259469279562338506, 4.68077281853310206106862164366, 5.46631430433200895385307270471, 6.33038895510137921413794483748, 7.51657625698454132307996153913, 8.656687122787811906829939170851, 9.015505740333110323366554758571, 10.30841298654330646085286594495

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