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Properties

Label 2-24e2-16.5-c3-0-4
Degree $2$
Conductor $576$
Sign $0.131 - 0.991i$
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  + (−14.8 − 14.8i)5-s + 10.4i·7-s + (−23.1 − 23.1i)11-s + (7.26 − 7.26i)13-s − 83.4·17-s + (81.4 − 81.4i)19-s − 86.6i·23-s + 314. i·25-s + (−37.1 + 37.1i)29-s − 251.·31-s + (154. − 154. i)35-s + (−102. − 102. i)37-s + 400. i·41-s + (332. + 332. i)43-s + 192.·47-s + ⋯
L(s)  = 1  + (−1.32 − 1.32i)5-s + 0.562i·7-s + (−0.634 − 0.634i)11-s + (0.154 − 0.154i)13-s − 1.19·17-s + (0.983 − 0.983i)19-s − 0.785i·23-s + 2.51i·25-s + (−0.238 + 0.238i)29-s − 1.45·31-s + (0.745 − 0.745i)35-s + (−0.454 − 0.454i)37-s + 1.52i·41-s + (1.18 + 1.18i)43-s + 0.597·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.131 - 0.991i$
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: \((2,\ 576,\ (\ :3/2),\ 0.131 - 0.991i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4046374700\)
\(L(\frac12)\) \(\approx\) \(0.4046374700\)
\(L(\frac{5}{2})\) 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 \( 1 + (14.8 + 14.8i)T + 125iT^{2} \)
7 \( 1 - 10.4iT - 343T^{2} \)
11 \( 1 + (23.1 + 23.1i)T + 1.33e3iT^{2} \)
13 \( 1 + (-7.26 + 7.26i)T - 2.19e3iT^{2} \)
17 \( 1 + 83.4T + 4.91e3T^{2} \)
19 \( 1 + (-81.4 + 81.4i)T - 6.85e3iT^{2} \)
23 \( 1 + 86.6iT - 1.21e4T^{2} \)
29 \( 1 + (37.1 - 37.1i)T - 2.43e4iT^{2} \)
31 \( 1 + 251.T + 2.97e4T^{2} \)
37 \( 1 + (102. + 102. i)T + 5.06e4iT^{2} \)
41 \( 1 - 400. iT - 6.89e4T^{2} \)
43 \( 1 + (-332. - 332. i)T + 7.95e4iT^{2} \)
47 \( 1 - 192.T + 1.03e5T^{2} \)
53 \( 1 + (-88.7 - 88.7i)T + 1.48e5iT^{2} \)
59 \( 1 + (-528. - 528. i)T + 2.05e5iT^{2} \)
61 \( 1 + (131. - 131. i)T - 2.26e5iT^{2} \)
67 \( 1 + (480. - 480. i)T - 3.00e5iT^{2} \)
71 \( 1 + 391. iT - 3.57e5T^{2} \)
73 \( 1 - 292. iT - 3.89e5T^{2} \)
79 \( 1 - 805.T + 4.93e5T^{2} \)
83 \( 1 + (-232. + 232. i)T - 5.71e5iT^{2} \)
89 \( 1 - 143. iT - 7.04e5T^{2} \)
97 \( 1 + 733.T + 9.12e5T^{2} \)
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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

−10.82297084558058589499556944537, −9.188727305935811374127346677884, −8.843595256431772110791699353848, −7.981843632382935424460335961203, −7.16619072667783072987134856220, −5.70160852105773187495847516160, −4.88791362986332858256454598034, −4.00188604362537439741870639415, −2.72468387951500410902955523201, −0.932905480454082944371452216686, 0.15215878269698994991594086031, 2.18935901918078664033237283951, 3.52727549574020553234183393942, 4.08275288167732531469315438554, 5.50127469109104233472890808234, 6.90851586552018607339934104290, 7.32297468905281764649496136766, 8.041502803809797147082080880421, 9.300085723909117257978973846687, 10.48441979177339244021192097912

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