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Properties

Label 2-24e2-9.5-c2-0-21
Degree $2$
Conductor $576$
Sign $0.804 - 0.594i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.18 + 2.05i)3-s + (2.05 + 1.18i)5-s + (4.05 + 7.02i)7-s + (0.558 − 8.98i)9-s + (17.6 − 10.1i)11-s + (3.05 − 5.29i)13-s + (−6.94 + 1.63i)15-s − 17.9i·17-s − 9.11·19-s + (−23.3 − 7.02i)21-s + (29.0 + 16.7i)23-s + (−9.67 − 16.7i)25-s + (17.2 + 20.7i)27-s + (−14.4 + 8.31i)29-s + (11.1 − 19.3i)31-s + ⋯
L(s)  = 1  + (−0.728 + 0.684i)3-s + (0.411 + 0.237i)5-s + (0.579 + 1.00i)7-s + (0.0620 − 0.998i)9-s + (1.60 − 0.924i)11-s + (0.235 − 0.407i)13-s + (−0.462 + 0.108i)15-s − 1.05i·17-s − 0.479·19-s + (−1.11 − 0.334i)21-s + (1.26 + 0.729i)23-s + (−0.387 − 0.670i)25-s + (0.638 + 0.769i)27-s + (−0.496 + 0.286i)29-s + (0.360 − 0.624i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.804 - 0.594i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.804 - 0.594i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.795780872\)
\(L(\frac12)\) \(\approx\) \(1.795780872\)
\(L(2)\) 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 \( 1 + (2.18 - 2.05i)T \)
good5 \( 1 + (-2.05 - 1.18i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-4.05 - 7.02i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-17.6 + 10.1i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-3.05 + 5.29i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 17.9iT - 289T^{2} \)
19 \( 1 + 9.11T + 361T^{2} \)
23 \( 1 + (-29.0 - 16.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (14.4 - 8.31i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-11.1 + 19.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 50.4T + 1.36e3T^{2} \)
41 \( 1 + (-29.9 - 17.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-11.5 - 19.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (33.1 - 19.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 19.0iT - 2.80e3T^{2} \)
59 \( 1 + (-2.96 - 1.71i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (23.1 + 40.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (3.14 - 5.45i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 35.9iT - 5.04e3T^{2} \)
73 \( 1 - 47.3T + 5.32e3T^{2} \)
79 \( 1 + (-42.2 - 73.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-33.1 + 19.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 143. iT - 7.92e3T^{2} \)
97 \( 1 + (40.3 + 69.9i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.86749222539001667483346404554, −9.391608588706294020791455996269, −9.311822350662611281208391148066, −8.110431972562570627202861103110, −6.62969788338621152845540357883, −5.97511816566812498630355170854, −5.16660999132995258174730359911, −4.04125400605622598471542647018, −2.80358446819467384618597747133, −1.04830864823181602116335778468, 1.09481984422210633427900701050, 1.87620493192179352172749446063, 4.02460555399002153083339821574, 4.74477587192890412294359569831, 6.05778431979703888929786292872, 6.78895396983433894587846263242, 7.51357217833080912680506073067, 8.639041273766701099807555873286, 9.606536750878993593387996238734, 10.66198430104848749874057711301

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