Properties

Label 2-24e2-36.31-c2-0-12
Degree $2$
Conductor $576$
Sign $0.866 - 0.499i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.262 − 2.98i)3-s + (−1.10 + 1.90i)5-s + (−7.23 + 4.17i)7-s + (−8.86 − 1.56i)9-s + (−4.54 + 2.62i)11-s + (7.37 − 12.7i)13-s + (5.40 + 3.79i)15-s + 28.2·17-s + 19.1i·19-s + (10.5 + 22.7i)21-s + (−3.16 − 1.82i)23-s + (10.0 + 17.4i)25-s + (−7.00 + 26.0i)27-s + (12.3 + 21.3i)29-s + (32.9 + 19.0i)31-s + ⋯
L(s)  = 1  + (0.0874 − 0.996i)3-s + (−0.220 + 0.381i)5-s + (−1.03 + 0.597i)7-s + (−0.984 − 0.174i)9-s + (−0.413 + 0.238i)11-s + (0.567 − 0.982i)13-s + (0.360 + 0.252i)15-s + 1.66·17-s + 1.00i·19-s + (0.504 + 1.08i)21-s + (−0.137 − 0.0794i)23-s + (0.403 + 0.698i)25-s + (−0.259 + 0.965i)27-s + (0.425 + 0.736i)29-s + (1.06 + 0.613i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.866 - 0.499i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.866 - 0.499i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.294753988\)
\(L(\frac12)\) \(\approx\) \(1.294753988\)
\(L(2)\) 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 \( 1 + (-0.262 + 2.98i)T \)
good5 \( 1 + (1.10 - 1.90i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (7.23 - 4.17i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (4.54 - 2.62i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-7.37 + 12.7i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 28.2T + 289T^{2} \)
19 \( 1 - 19.1iT - 361T^{2} \)
23 \( 1 + (3.16 + 1.82i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-12.3 - 21.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-32.9 - 19.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 4.21T + 1.36e3T^{2} \)
41 \( 1 + (9.92 - 17.1i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-20.1 + 11.6i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (25.8 - 14.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 32.1T + 2.80e3T^{2} \)
59 \( 1 + (7.96 + 4.59i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-40.8 - 70.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6.86 - 3.96i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 62.9iT - 5.04e3T^{2} \)
73 \( 1 - 33.3T + 5.32e3T^{2} \)
79 \( 1 + (53.7 - 31.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (103. - 59.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 107.T + 7.92e3T^{2} \)
97 \( 1 + (-1.78 - 3.09i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.50677531066828747022569412517, −9.814840296176898537095206838755, −8.592987930979136284608560808497, −7.901418612617048332843820634947, −7.02708532383840763649665100840, −6.05255059168077623358648124413, −5.42400519103211018047303273120, −3.39020293306135367878578921000, −2.84831356543961314235016088756, −1.15353387266312445358071011527, 0.57313952389063208341393508168, 2.81588031343660404846401460982, 3.79112167312614413279292302333, 4.63201112998586215931207813110, 5.78383419276047289826931575105, 6.73123746760304110026524535953, 7.983122836410068373365952077429, 8.797771939536551349162564400213, 9.780362185783525762414872109222, 10.16745703288228521312829219730

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