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Properties

Label 2-864-9.5-c2-0-14
Degree $2$
Conductor $864$
Sign $0.00681 + 0.999i$
Analytic cond. $23.5422$
Root an. cond. $4.85204$
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  + (−5.41 − 3.12i)5-s + (3.74 + 6.49i)7-s + (6.17 − 3.56i)11-s + (−0.888 + 1.53i)13-s − 14.7i·17-s − 19.9·19-s + (20.8 + 12.0i)23-s + (7.03 + 12.1i)25-s + (−40.1 + 23.1i)29-s + (14.1 − 24.5i)31-s − 46.8i·35-s + 63.0·37-s + (28.0 + 16.1i)41-s + (−38.8 − 67.2i)43-s + (38.4 − 22.2i)47-s + ⋯
L(s)  = 1  + (−1.08 − 0.625i)5-s + (0.535 + 0.927i)7-s + (0.561 − 0.324i)11-s + (−0.0683 + 0.118i)13-s − 0.869i·17-s − 1.04·19-s + (0.907 + 0.523i)23-s + (0.281 + 0.487i)25-s + (−1.38 + 0.798i)29-s + (0.456 − 0.790i)31-s − 1.33i·35-s + 1.70·37-s + (0.683 + 0.394i)41-s + (−0.902 − 1.56i)43-s + (0.818 − 0.472i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00681 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00681 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.00681 + 0.999i$
Analytic conductor: \(23.5422\)
Root analytic conductor: \(4.85204\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1),\ 0.00681 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.153674500\)
\(L(\frac12)\) \(\approx\) \(1.153674500\)
\(L(2)\) 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 \( 1 \)
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

−9.401189470109292628820687261839, −8.937554204409081609251800949589, −8.116772085409392942076304542715, −7.39036863734613415949918597083, −6.23696632093471127446093203369, −5.18518665973496425989621451283, −4.41825035423542562569148223015, −3.37082204931410259401212129891, −1.97880239174633552939571573617, −0.43115209277748592674697589857, 1.19557973065638955832278293003, 2.78995674984954172418860747262, 4.14898601425745734951311813554, 4.31425002238231150547035620048, 5.97787443792533052069671039034, 6.93329730773723431503799343600, 7.60966810194426634776850898290, 8.271943078676271022435783222724, 9.327159091442859742338551437471, 10.45069650039876994834128327120

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