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Properties

Label 2-864-36.23-c1-0-6
Degree $2$
Conductor $864$
Sign $0.811 - 0.584i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.68 + 0.971i)5-s + (2.61 − 1.50i)7-s + (2.20 + 3.81i)11-s + (−2.65 + 4.60i)13-s + 4.16i·17-s − 4.66i·19-s + (1.30 − 2.26i)23-s + (−0.613 − 1.06i)25-s + (−1.13 + 0.655i)29-s + (0.0648 + 0.0374i)31-s + 5.86·35-s + 8.43·37-s + (−8.16 − 4.71i)41-s + (5.06 − 2.92i)43-s + (2.40 + 4.16i)47-s + ⋯
L(s)  = 1  + (0.752 + 0.434i)5-s + (0.988 − 0.570i)7-s + (0.664 + 1.15i)11-s + (−0.737 + 1.27i)13-s + 1.00i·17-s − 1.06i·19-s + (0.272 − 0.471i)23-s + (−0.122 − 0.212i)25-s + (−0.210 + 0.121i)29-s + (0.0116 + 0.00672i)31-s + 0.991·35-s + 1.38·37-s + (−1.27 − 0.735i)41-s + (0.772 − 0.446i)43-s + (0.350 + 0.607i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.811 - 0.584i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.811 - 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87939 + 0.606822i\)
\(L(\frac12)\) \(\approx\) \(1.87939 + 0.606822i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.971i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.61 + 1.50i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.20 - 3.81i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.65 - 4.60i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.16iT - 17T^{2} \)
19 \( 1 + 4.66iT - 19T^{2} \)
23 \( 1 + (-1.30 + 2.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.13 - 0.655i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.0648 - 0.0374i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.43T + 37T^{2} \)
41 \( 1 + (8.16 + 4.71i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.06 + 2.92i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.40 - 4.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.96iT - 53T^{2} \)
59 \( 1 + (1.74 - 3.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.32 - 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.38 + 1.37i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.910T + 71T^{2} \)
73 \( 1 - 6.86T + 73T^{2} \)
79 \( 1 + (-10.8 + 6.28i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.61 + 14.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.96iT - 89T^{2} \)
97 \( 1 + (9.03 + 15.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.24339379491342800182182624259, −9.462177637057513965903045455308, −8.712125147118362797149289604050, −7.44940284725482051041616629806, −6.93709449622409934590469779327, −6.00902512917119041436023875705, −4.65364004537000205767857182199, −4.23180340865107049787717246138, −2.42423673896309010824122390634, −1.58791220282600512945658551011, 1.09161177236160635990807837007, 2.38394952242419834417358769055, 3.59618034568178791556112435362, 5.16290369898685910731782031936, 5.41555594821738231213938914831, 6.46128407800004607750679253344, 7.82958861428003758122859515173, 8.295207984131670539640515464565, 9.349362111114605921503067236077, 9.866198865053123429820136570369

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