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Properties

Label 2-24e2-9.5-c2-0-1
Degree $2$
Conductor $576$
Sign $-0.987 - 0.160i$
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  + (−2.81 + 1.04i)3-s + (−5.41 − 3.12i)5-s + (3.74 + 6.49i)7-s + (6.81 − 5.87i)9-s + (6.17 − 3.56i)11-s + (0.888 − 1.53i)13-s + (18.4 + 3.13i)15-s + 14.7i·17-s + 19.9·19-s + (−17.3 − 14.3i)21-s + (−20.8 − 12.0i)23-s + (7.03 + 12.1i)25-s + (−13.0 + 23.6i)27-s + (−40.1 + 23.1i)29-s + (14.1 − 24.5i)31-s + ⋯
L(s)  = 1  + (−0.937 + 0.348i)3-s + (−1.08 − 0.625i)5-s + (0.535 + 0.927i)7-s + (0.757 − 0.653i)9-s + (0.561 − 0.324i)11-s + (0.0683 − 0.118i)13-s + (1.23 + 0.208i)15-s + 0.869i·17-s + 1.04·19-s + (−0.825 − 0.682i)21-s + (−0.907 − 0.523i)23-s + (0.281 + 0.487i)25-s + (−0.482 + 0.876i)27-s + (−1.38 + 0.798i)29-s + (0.456 − 0.790i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.987 - 0.160i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.987 - 0.160i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2024679700\)
\(L(\frac12)\) \(\approx\) \(0.2024679700\)
\(L(2)\) 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 \( 1 + (2.81 - 1.04i)T \)
good5 \( 1 + (5.41 + 3.12i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.74 - 6.49i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-6.17 + 3.56i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.888 + 1.53i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 14.7iT - 289T^{2} \)
19 \( 1 - 19.9T + 361T^{2} \)
23 \( 1 + (20.8 + 12.0i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (40.1 - 23.1i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-14.1 + 24.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 63.0T + 1.36e3T^{2} \)
41 \( 1 + (28.0 + 16.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-38.8 - 67.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (38.4 - 22.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 42.2iT - 2.80e3T^{2} \)
59 \( 1 + (93.8 + 54.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-25.3 - 43.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (56.9 - 98.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 85.2iT - 5.04e3T^{2} \)
73 \( 1 + 94.5T + 5.32e3T^{2} \)
79 \( 1 + (35.6 + 61.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-94.9 + 54.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 29.6iT - 7.92e3T^{2} \)
97 \( 1 + (62.4 + 108. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.23038876617742170554896576636, −10.15601159281159223668638924999, −9.091794020758475097531842278658, −8.363117633283856556490256693813, −7.43651658403483583956469315602, −6.16228641742895270310575589782, −5.38187921963791235802494702828, −4.43475591620572259611619007663, −3.53656609021603290724957528832, −1.47012325646115806450238781220, 0.094740129031897623125009362265, 1.55802964520017312845464003610, 3.50910230245924532332737517748, 4.38656299979700848505801386282, 5.42639913087918791811907663837, 6.73129767610881790611369408390, 7.39079085089815876063545077260, 7.83512309922610633944630482580, 9.375086603006203245702996171579, 10.43535176810373170930731325037

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