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Properties

Label 2-24e2-16.5-c3-0-22
Degree $2$
Conductor $576$
Sign $0.287 + 0.957i$
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  + (5.80 + 5.80i)5-s − 3.41i·7-s + (−24.3 − 24.3i)11-s + (−23.6 + 23.6i)13-s − 3.82·17-s + (80.9 − 80.9i)19-s − 133. i·23-s − 57.6i·25-s + (−55.9 + 55.9i)29-s + 87.0·31-s + (19.8 − 19.8i)35-s + (−151. − 151. i)37-s + 228. i·41-s + (−209. − 209. i)43-s + 91.1·47-s + ⋯
L(s)  = 1  + (0.519 + 0.519i)5-s − 0.184i·7-s + (−0.668 − 0.668i)11-s + (−0.504 + 0.504i)13-s − 0.0546·17-s + (0.977 − 0.977i)19-s − 1.21i·23-s − 0.460i·25-s + (−0.358 + 0.358i)29-s + 0.504·31-s + (0.0957 − 0.0957i)35-s + (−0.674 − 0.674i)37-s + 0.872i·41-s + (−0.741 − 0.741i)43-s + 0.282·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.287 + 0.957i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ 0.287 + 0.957i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.610772215\)
\(L(\frac12)\) \(\approx\) \(1.610772215\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.80 - 5.80i)T + 125iT^{2} \)
7 \( 1 + 3.41iT - 343T^{2} \)
11 \( 1 + (24.3 + 24.3i)T + 1.33e3iT^{2} \)
13 \( 1 + (23.6 - 23.6i)T - 2.19e3iT^{2} \)
17 \( 1 + 3.82T + 4.91e3T^{2} \)
19 \( 1 + (-80.9 + 80.9i)T - 6.85e3iT^{2} \)
23 \( 1 + 133. iT - 1.21e4T^{2} \)
29 \( 1 + (55.9 - 55.9i)T - 2.43e4iT^{2} \)
31 \( 1 - 87.0T + 2.97e4T^{2} \)
37 \( 1 + (151. + 151. i)T + 5.06e4iT^{2} \)
41 \( 1 - 228. iT - 6.89e4T^{2} \)
43 \( 1 + (209. + 209. i)T + 7.95e4iT^{2} \)
47 \( 1 - 91.1T + 1.03e5T^{2} \)
53 \( 1 + (-484. - 484. i)T + 1.48e5iT^{2} \)
59 \( 1 + (161. + 161. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-546. + 546. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-270. + 270. i)T - 3.00e5iT^{2} \)
71 \( 1 + 648. iT - 3.57e5T^{2} \)
73 \( 1 + 1.06e3iT - 3.89e5T^{2} \)
79 \( 1 + 85.2T + 4.93e5T^{2} \)
83 \( 1 + (698. - 698. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.39e3iT - 7.04e5T^{2} \)
97 \( 1 + 390.T + 9.12e5T^{2} \)
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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

−10.25808104192250799442167673506, −9.313295125743862249111885448609, −8.429440603668413736242896795996, −7.32545202535930577866279370058, −6.58152814008101966585090129371, −5.54311092473173244179820812234, −4.56364051932642477553061447117, −3.14379626315691653797605791084, −2.22668347087837162520405243757, −0.49395963064582167655701901693, 1.27398967194494536469669019909, 2.49292537151627787524441054207, 3.81354896762265482044053036821, 5.27756162644117819917208395330, 5.52585489464650940866821993877, 7.04120774114485321946082484402, 7.80985342049693170019651947677, 8.777002057880840108174132528984, 9.878694254056242778215995884577, 10.09567091338893852578000653510

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