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Properties

Label 2-864-72.13-c1-0-8
Degree $2$
Conductor $864$
Sign $-0.0339 + 0.999i$
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  + (−1.97 − 1.14i)5-s + (0.907 + 1.57i)7-s + (4.24 − 2.44i)11-s + (−4.00 − 2.31i)13-s − 1.92·17-s − 2.12i·19-s + (−1.15 + 2.00i)23-s + (0.101 + 0.175i)25-s + (3.16 − 1.82i)29-s + (2.65 − 4.60i)31-s − 4.14i·35-s − 7.98i·37-s + (2.36 − 4.09i)41-s + (−2.20 + 1.27i)43-s + (−2.02 − 3.49i)47-s + ⋯
L(s)  = 1  + (−0.883 − 0.510i)5-s + (0.343 + 0.594i)7-s + (1.27 − 0.738i)11-s + (−1.11 − 0.641i)13-s − 0.467·17-s − 0.488i·19-s + (−0.241 + 0.418i)23-s + (0.0203 + 0.0351i)25-s + (0.587 − 0.339i)29-s + (0.477 − 0.826i)31-s − 0.700i·35-s − 1.31i·37-s + (0.368 − 0.639i)41-s + (−0.336 + 0.194i)43-s + (−0.294 − 0.510i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0339 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0339 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.0339 + 0.999i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.0339 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753208 - 0.779200i\)
\(L(\frac12)\) \(\approx\) \(0.753208 - 0.779200i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.97 + 1.14i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.907 - 1.57i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.24 + 2.44i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.00 + 2.31i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.92T + 17T^{2} \)
19 \( 1 + 2.12iT - 19T^{2} \)
23 \( 1 + (1.15 - 2.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.16 + 1.82i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.65 + 4.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.98iT - 37T^{2} \)
41 \( 1 + (-2.36 + 4.09i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.20 - 1.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.02 + 3.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.95iT - 53T^{2} \)
59 \( 1 + (3.05 + 1.76i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.71 - 0.991i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.72 + 4.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + (-4.97 - 8.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.12 - 1.80i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.49T + 89T^{2} \)
97 \( 1 + (-6.99 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.803755786444429672879118076330, −8.990328126279343773296625455615, −8.316451443904873303539891450101, −7.54455209860222146957232729198, −6.49648820042369068235114378262, −5.46805541563935615301144324016, −4.52437371560592616279328702982, −3.62929126330577397142645263408, −2.28388178163278665223417458559, −0.54771510000560761205454423353, 1.53275785569826115772045141429, 3.03187742203160023374863767289, 4.26429987599192991234923589333, 4.63642629186678952180671448493, 6.34408041277561866184393763644, 7.04870943694992333232019051817, 7.64558037129618182631365507860, 8.668634567979599656059477234938, 9.618020922745342162454836103757, 10.37495478144654395175170746304

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