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Properties

Label 2-24e2-48.11-c3-0-3
Degree $2$
Conductor $576$
Sign $-0.575 - 0.817i$
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  + (−2.40 + 2.40i)5-s + 11.7·7-s + (−34.7 − 34.7i)11-s + (−3.17 + 3.17i)13-s + 98.0i·17-s + (15.9 + 15.9i)19-s − 69.6i·23-s + 113. i·25-s + (15.9 + 15.9i)29-s + 121. i·31-s + (−28.2 + 28.2i)35-s + (−37.0 − 37.0i)37-s − 59.3·41-s + (241. − 241. i)43-s − 395.·47-s + ⋯
L(s)  = 1  + (−0.215 + 0.215i)5-s + 0.632·7-s + (−0.952 − 0.952i)11-s + (−0.0678 + 0.0678i)13-s + 1.39i·17-s + (0.192 + 0.192i)19-s − 0.631i·23-s + 0.907i·25-s + (0.102 + 0.102i)29-s + 0.702i·31-s + (−0.136 + 0.136i)35-s + (−0.164 − 0.164i)37-s − 0.225·41-s + (0.857 − 0.857i)43-s − 1.22·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.575 - 0.817i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ -0.575 - 0.817i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9229532055\)
\(L(\frac12)\) \(\approx\) \(0.9229532055\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.40 - 2.40i)T - 125iT^{2} \)
7 \( 1 - 11.7T + 343T^{2} \)
11 \( 1 + (34.7 + 34.7i)T + 1.33e3iT^{2} \)
13 \( 1 + (3.17 - 3.17i)T - 2.19e3iT^{2} \)
17 \( 1 - 98.0iT - 4.91e3T^{2} \)
19 \( 1 + (-15.9 - 15.9i)T + 6.85e3iT^{2} \)
23 \( 1 + 69.6iT - 1.21e4T^{2} \)
29 \( 1 + (-15.9 - 15.9i)T + 2.43e4iT^{2} \)
31 \( 1 - 121. iT - 2.97e4T^{2} \)
37 \( 1 + (37.0 + 37.0i)T + 5.06e4iT^{2} \)
41 \( 1 + 59.3T + 6.89e4T^{2} \)
43 \( 1 + (-241. + 241. i)T - 7.95e4iT^{2} \)
47 \( 1 + 395.T + 1.03e5T^{2} \)
53 \( 1 + (458. - 458. i)T - 1.48e5iT^{2} \)
59 \( 1 + (257. + 257. i)T + 2.05e5iT^{2} \)
61 \( 1 + (373. - 373. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-648. - 648. i)T + 3.00e5iT^{2} \)
71 \( 1 - 787. iT - 3.57e5T^{2} \)
73 \( 1 - 1.07e3iT - 3.89e5T^{2} \)
79 \( 1 - 382. iT - 4.93e5T^{2} \)
83 \( 1 + (491. - 491. i)T - 5.71e5iT^{2} \)
89 \( 1 + 624.T + 7.04e5T^{2} \)
97 \( 1 - 1.66e3T + 9.12e5T^{2} \)
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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

−10.85301146335238486188803954850, −9.876711640337683340814040861724, −8.575510315274121703625397812620, −8.144549287455405647202256431914, −7.12147738961219628964013459314, −5.97581604290441118315515885633, −5.13987521686405830759865161301, −3.92891552702153387926699156161, −2.83177443521578552550992491505, −1.40897609520976445705861930041, 0.27153939511387907623363876714, 1.88663445147195212154837218863, 3.06874862141766250497885523918, 4.65049625002367267489457524594, 5.04044038601626259194446874533, 6.41887889504706215199809217122, 7.61660841014596919456774142616, 7.959653692365072848022981961096, 9.272953027926231681385072503785, 9.918427515473480887132738496638

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