Properties

Label 2-24e2-16.5-c3-0-3
Degree $2$
Conductor $576$
Sign $-0.978 + 0.206i$
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  + (0.596 + 0.596i)5-s + 29.0i·7-s + (12.1 + 12.1i)11-s + (−48.5 + 48.5i)13-s − 86.7·17-s + (54.8 − 54.8i)19-s − 70.2i·23-s − 124. i·25-s + (−63.4 + 63.4i)29-s + 8.86·31-s + (−17.3 + 17.3i)35-s + (−21.7 − 21.7i)37-s − 153. i·41-s + (120. + 120. i)43-s − 99.9·47-s + ⋯
L(s)  = 1  + (0.0533 + 0.0533i)5-s + 1.57i·7-s + (0.332 + 0.332i)11-s + (−1.03 + 1.03i)13-s − 1.23·17-s + (0.662 − 0.662i)19-s − 0.636i·23-s − 0.994i·25-s + (−0.405 + 0.405i)29-s + 0.0513·31-s + (−0.0838 + 0.0838i)35-s + (−0.0964 − 0.0964i)37-s − 0.583i·41-s + (0.428 + 0.428i)43-s − 0.310·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.978 + 0.206i$
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: \((2,\ 576,\ (\ :3/2),\ -0.978 + 0.206i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4733413119\)
\(L(\frac12)\) \(\approx\) \(0.4733413119\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.596 - 0.596i)T + 125iT^{2} \)
7 \( 1 - 29.0iT - 343T^{2} \)
11 \( 1 + (-12.1 - 12.1i)T + 1.33e3iT^{2} \)
13 \( 1 + (48.5 - 48.5i)T - 2.19e3iT^{2} \)
17 \( 1 + 86.7T + 4.91e3T^{2} \)
19 \( 1 + (-54.8 + 54.8i)T - 6.85e3iT^{2} \)
23 \( 1 + 70.2iT - 1.21e4T^{2} \)
29 \( 1 + (63.4 - 63.4i)T - 2.43e4iT^{2} \)
31 \( 1 - 8.86T + 2.97e4T^{2} \)
37 \( 1 + (21.7 + 21.7i)T + 5.06e4iT^{2} \)
41 \( 1 + 153. iT - 6.89e4T^{2} \)
43 \( 1 + (-120. - 120. i)T + 7.95e4iT^{2} \)
47 \( 1 + 99.9T + 1.03e5T^{2} \)
53 \( 1 + (389. + 389. i)T + 1.48e5iT^{2} \)
59 \( 1 + (324. + 324. i)T + 2.05e5iT^{2} \)
61 \( 1 + (0.339 - 0.339i)T - 2.26e5iT^{2} \)
67 \( 1 + (565. - 565. i)T - 3.00e5iT^{2} \)
71 \( 1 - 419. iT - 3.57e5T^{2} \)
73 \( 1 + 374. iT - 3.89e5T^{2} \)
79 \( 1 - 705.T + 4.93e5T^{2} \)
83 \( 1 + (947. - 947. i)T - 5.71e5iT^{2} \)
89 \( 1 + 4.72iT - 7.04e5T^{2} \)
97 \( 1 - 379.T + 9.12e5T^{2} \)
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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

−10.90647338199989488279847623323, −9.593795485341615356465971292085, −9.165027208085639894638599176703, −8.328668969449671544020612500630, −7.03653788826206833076382255851, −6.34738137776636664925651830341, −5.17564271712802109514930116070, −4.38004643541493700768853934196, −2.71544365626811988175121895301, −1.98300546976644516659445085855, 0.13581468507461638303729159335, 1.41451703920941481485613579523, 3.08522178207019048314004738374, 4.08839724576060602896219454831, 5.09308270395957164399455537116, 6.25375197517412538361876269401, 7.39702860419189014664190604308, 7.74513683163713217631685652672, 9.114349158646396303980213202687, 9.929774677559431678165602365786

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