Properties

Label 2-24e2-8.5-c3-0-24
Degree $2$
Conductor $576$
Sign $-0.707 + 0.707i$
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  + 18i·11-s − 90·17-s − 106i·19-s + 125·25-s − 522·41-s − 290i·43-s − 343·49-s − 846i·59-s + 70i·67-s − 430·73-s − 1.35e3i·83-s − 1.02e3·89-s − 1.91e3·97-s − 1.71e3i·107-s + 270·113-s + ⋯
L(s)  = 1  + 0.493i·11-s − 1.28·17-s − 1.27i·19-s + 25-s − 1.98·41-s − 1.02i·43-s − 49-s − 1.86i·59-s + 0.127i·67-s − 0.689·73-s − 1.78i·83-s − 1.22·89-s − 1.99·97-s − 1.54i·107-s + 0.224·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.707 + 0.707i$
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: \((2,\ 576,\ (\ :3/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6923090619\)
\(L(\frac12)\) \(\approx\) \(0.6923090619\)
\(L(\frac{5}{2})\) 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 \( 1 - 125T^{2} \)
7 \( 1 + 343T^{2} \)
11 \( 1 - 18iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 90T + 4.91e3T^{2} \)
19 \( 1 + 106iT - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 + 522T + 6.89e4T^{2} \)
43 \( 1 + 290iT - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 846iT - 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 70iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 430T + 3.89e5T^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 + 1.35e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + 1.91e3T + 9.12e5T^{2} \)
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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

−9.990546827521816250209878672378, −9.050586010279375456661197467882, −8.375186935866125219423757538266, −7.08345020592723827539044656372, −6.59067541847462047433441894719, −5.17684174673876907358549479866, −4.42637463352687941659332614256, −3.06510397444371321587764166741, −1.86993591028199744922661392953, −0.20003468293914237381110268328, 1.44723192460365610634002241786, 2.84269744984927898170892811505, 4.00340861063318723668598869248, 5.08308260030683094613814473702, 6.16734807862017247304645257915, 6.97289569395717397899705866122, 8.147892752543172415150220471379, 8.783472809410815713234616394584, 9.819188016137967151442135240238, 10.67016050514561331154485571560

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