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Properties

Label 2-567-21.20-c3-0-87
Degree $2$
Conductor $567$
Sign $-0.974 + 0.225i$
Analytic cond. $33.4540$
Root an. cond. $5.78395$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.99i·2-s − 0.957·4-s + 15.6·5-s + (−4.16 − 18.0i)7-s − 21.0i·8-s − 46.6i·10-s − 53.6i·11-s − 33.6i·13-s + (−54.0 + 12.4i)14-s − 70.7·16-s + 43.5·17-s + 32.2i·19-s − 14.9·20-s − 160.·22-s + 149. i·23-s + ⋯
L(s)  = 1  − 1.05i·2-s − 0.119·4-s + 1.39·5-s + (−0.225 − 0.974i)7-s − 0.931i·8-s − 1.47i·10-s − 1.47i·11-s − 0.717i·13-s + (−1.03 + 0.238i)14-s − 1.10·16-s + 0.621·17-s + 0.389i·19-s − 0.166·20-s − 1.55·22-s + 1.35i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.974 + 0.225i$
Analytic conductor: \(33.4540\)
Root analytic conductor: \(5.78395\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :3/2),\ -0.974 + 0.225i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.647887484\)
\(L(\frac12)\) \(\approx\) \(2.647887484\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (4.16 + 18.0i)T \)
good2 \( 1 + 2.99iT - 8T^{2} \)
5 \( 1 - 15.6T + 125T^{2} \)
11 \( 1 + 53.6iT - 1.33e3T^{2} \)
13 \( 1 + 33.6iT - 2.19e3T^{2} \)
17 \( 1 - 43.5T + 4.91e3T^{2} \)
19 \( 1 - 32.2iT - 6.85e3T^{2} \)
23 \( 1 - 149. iT - 1.21e4T^{2} \)
29 \( 1 - 147. iT - 2.43e4T^{2} \)
31 \( 1 + 70.6iT - 2.97e4T^{2} \)
37 \( 1 - 40.3T + 5.06e4T^{2} \)
41 \( 1 + 358.T + 6.89e4T^{2} \)
43 \( 1 - 507.T + 7.95e4T^{2} \)
47 \( 1 + 456.T + 1.03e5T^{2} \)
53 \( 1 + 213. iT - 1.48e5T^{2} \)
59 \( 1 - 319.T + 2.05e5T^{2} \)
61 \( 1 + 350. iT - 2.26e5T^{2} \)
67 \( 1 - 578.T + 3.00e5T^{2} \)
71 \( 1 - 787. iT - 3.57e5T^{2} \)
73 \( 1 - 146. iT - 3.89e5T^{2} \)
79 \( 1 - 386.T + 4.93e5T^{2} \)
83 \( 1 - 197.T + 5.71e5T^{2} \)
89 \( 1 + 596.T + 7.04e5T^{2} \)
97 \( 1 + 729. iT - 9.12e5T^{2} \)
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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

−9.953955008314420969597695272281, −9.625437077722700582946712133434, −8.328442724166973742371069741242, −7.14405285111825517785802211612, −6.13060041796683435708410180430, −5.37063312595851702474419631784, −3.67320716375683219255605954234, −3.03015802259576189743962564527, −1.65366885859852774432973985907, −0.75666590009727528342631475575, 1.90617003586554361137058471351, 2.53557253890233074591141179133, 4.65665516013742100529955680713, 5.45167558355323613367858415360, 6.33756115948481676428539642458, 6.83390355638872881246195195658, 7.992838254148333978346488865632, 9.027817042077964272875391423574, 9.624033053936905624507779377707, 10.48798905338208359685168241234

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