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Properties

Label 2-24e2-16.11-c4-0-36
Degree $2$
Conductor $576$
Sign $-0.756 + 0.654i$
Analytic cond. $59.5410$
Root an. cond. $7.71628$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.04 + 8.04i)5-s + 49.8·7-s + (−84.2 + 84.2i)11-s + (19.4 − 19.4i)13-s − 437.·17-s + (−349. − 349. i)19-s − 404.·23-s − 495. i·25-s + (1.03e3 − 1.03e3i)29-s + 1.50e3i·31-s + (401. + 401. i)35-s + (−434. − 434. i)37-s − 696. i·41-s + (−917. + 917. i)43-s + 111. i·47-s + ⋯
L(s)  = 1  + (0.321 + 0.321i)5-s + 1.01·7-s + (−0.696 + 0.696i)11-s + (0.115 − 0.115i)13-s − 1.51·17-s + (−0.966 − 0.966i)19-s − 0.765·23-s − 0.792i·25-s + (1.22 − 1.22i)29-s + 1.56i·31-s + (0.327 + 0.327i)35-s + (−0.317 − 0.317i)37-s − 0.414i·41-s + (−0.496 + 0.496i)43-s + 0.0506i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.756 + 0.654i$
Analytic conductor: \(59.5410\)
Root analytic conductor: \(7.71628\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :2),\ -0.756 + 0.654i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5616285420\)
\(L(\frac12)\) \(\approx\) \(0.5616285420\)
\(L(3)\) 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 \( 1 + (-8.04 - 8.04i)T + 625iT^{2} \)
7 \( 1 - 49.8T + 2.40e3T^{2} \)
11 \( 1 + (84.2 - 84.2i)T - 1.46e4iT^{2} \)
13 \( 1 + (-19.4 + 19.4i)T - 2.85e4iT^{2} \)
17 \( 1 + 437.T + 8.35e4T^{2} \)
19 \( 1 + (349. + 349. i)T + 1.30e5iT^{2} \)
23 \( 1 + 404.T + 2.79e5T^{2} \)
29 \( 1 + (-1.03e3 + 1.03e3i)T - 7.07e5iT^{2} \)
31 \( 1 - 1.50e3iT - 9.23e5T^{2} \)
37 \( 1 + (434. + 434. i)T + 1.87e6iT^{2} \)
41 \( 1 + 696. iT - 2.82e6T^{2} \)
43 \( 1 + (917. - 917. i)T - 3.41e6iT^{2} \)
47 \( 1 - 111. iT - 4.87e6T^{2} \)
53 \( 1 + (1.04e3 + 1.04e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (1.71e3 - 1.71e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-3.71e3 + 3.71e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (-1.85e3 - 1.85e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 1.16e3T + 2.54e7T^{2} \)
73 \( 1 + 905. iT - 2.83e7T^{2} \)
79 \( 1 + 5.86e3iT - 3.89e7T^{2} \)
83 \( 1 + (7.56e3 + 7.56e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 6.43e3iT - 6.27e7T^{2} \)
97 \( 1 + 413.T + 8.85e7T^{2} \)
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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

−9.977912313879748689262820764219, −8.719392207499806648343354046115, −8.162311517808790497880957172028, −7.01093236147299514446288782916, −6.25727991961854222635184725565, −4.90577693376507567528908923968, −4.37103048524401804347760402656, −2.62155340717813596667811933822, −1.86716837007469290040590626732, −0.12850575407334338389119738232, 1.43493715591424103998498544924, 2.43858005789321661199654330833, 3.97280683028171245539320800327, 4.92034870500956523619880248965, 5.82267106859293668248253657503, 6.81749438735603311558875224665, 8.169205394246382294377123659718, 8.437894727519783333803551270676, 9.560748817472839138308478402424, 10.65728138066299832676476110448

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