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Properties

Label 2-864-8.3-c2-0-10
Degree $2$
Conductor $864$
Sign $-0.404 - 0.914i$
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  + 5.51i·5-s + 11.2i·7-s + 17.9·11-s − 5.65i·13-s − 9.53·17-s + 33.1·19-s + 19.3i·23-s − 5.42·25-s + 29.8i·29-s − 6.52i·31-s − 61.8·35-s − 33.9i·37-s − 56.8·41-s − 19.1·43-s + 30.9i·47-s + ⋯
L(s)  = 1  + 1.10i·5-s + 1.60i·7-s + 1.62·11-s − 0.435i·13-s − 0.561·17-s + 1.74·19-s + 0.843i·23-s − 0.216·25-s + 1.02i·29-s − 0.210i·31-s − 1.76·35-s − 0.916i·37-s − 1.38·41-s − 0.444·43-s + 0.659i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.404 - 0.914i$
Analytic conductor: \(23.5422\)
Root analytic conductor: \(4.85204\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1),\ -0.404 - 0.914i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.961341483\)
\(L(\frac12)\) \(\approx\) \(1.961341483\)
\(L(2)\) 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 \( 1 - 5.51iT - 25T^{2} \)
7 \( 1 - 11.2iT - 49T^{2} \)
11 \( 1 - 17.9T + 121T^{2} \)
13 \( 1 + 5.65iT - 169T^{2} \)
17 \( 1 + 9.53T + 289T^{2} \)
19 \( 1 - 33.1T + 361T^{2} \)
23 \( 1 - 19.3iT - 529T^{2} \)
29 \( 1 - 29.8iT - 841T^{2} \)
31 \( 1 + 6.52iT - 961T^{2} \)
37 \( 1 + 33.9iT - 1.36e3T^{2} \)
41 \( 1 + 56.8T + 1.68e3T^{2} \)
43 \( 1 + 19.1T + 1.84e3T^{2} \)
47 \( 1 - 30.9iT - 2.20e3T^{2} \)
53 \( 1 - 11.2iT - 2.80e3T^{2} \)
59 \( 1 + 10.2T + 3.48e3T^{2} \)
61 \( 1 + 3.18iT - 3.72e3T^{2} \)
67 \( 1 + 13.1T + 4.48e3T^{2} \)
71 \( 1 - 103. iT - 5.04e3T^{2} \)
73 \( 1 + 21.3T + 5.32e3T^{2} \)
79 \( 1 + 134. iT - 6.24e3T^{2} \)
83 \( 1 + 56.2T + 6.88e3T^{2} \)
89 \( 1 - 114.T + 7.92e3T^{2} \)
97 \( 1 - 126.T + 9.40e3T^{2} \)
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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

−10.15556979152099560957922225310, −9.256342165727423955070285732364, −8.818368540485287270854050353638, −7.55057421707256068528816720481, −6.74278023296002611620951351036, −5.95290285900909589308150019379, −5.11318082639149473355029817965, −3.57240531356771903922419526954, −2.86721499775033891523675216079, −1.59402561745978999632947830235, 0.71606306320084245602394898629, 1.50950985262360956956182925665, 3.50325472556831150392057739603, 4.30730020836012376544817278347, 4.99672947071806884465965097342, 6.44341989131882185414107202816, 7.03661415272060595329809282134, 8.056929086629941245735987263772, 8.943008517492191030090799597281, 9.633391917008517632432260130543

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