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Properties

Label 2-567-63.25-c1-0-11
Degree $2$
Conductor $567$
Sign $0.888 - 0.458i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
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  − 2.44·2-s + 3.99·4-s + (1.22 − 2.12i)5-s + (2.5 + 0.866i)7-s − 4.89·8-s + (−2.99 + 5.19i)10-s + (2.44 + 4.24i)11-s + (2 + 3.46i)13-s + (−6.12 − 2.12i)14-s + 3.99·16-s + (−1.22 + 2.12i)17-s + (0.5 + 0.866i)19-s + (4.89 − 8.48i)20-s + (−5.99 − 10.3i)22-s + (1.22 − 2.12i)23-s + ⋯
L(s)  = 1  − 1.73·2-s + 1.99·4-s + (0.547 − 0.948i)5-s + (0.944 + 0.327i)7-s − 1.73·8-s + (−0.948 + 1.64i)10-s + (0.738 + 1.27i)11-s + (0.554 + 0.960i)13-s + (−1.63 − 0.566i)14-s + 0.999·16-s + (−0.297 + 0.514i)17-s + (0.114 + 0.198i)19-s + (1.09 − 1.89i)20-s + (−1.27 − 2.21i)22-s + (0.255 − 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.796261 + 0.193093i\)
\(L(\frac12)\) \(\approx\) \(0.796261 + 0.193093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.5 - 0.866i)T \)
good2 \( 1 + 2.44T + 2T^{2} \)
5 \( 1 + (-1.22 + 2.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.44 - 4.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.22 - 2.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.22 + 2.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.67 - 6.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.67 - 6.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.44T + 47T^{2} \)
53 \( 1 + (1.22 - 2.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.79T + 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + (-7.34 + 12.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.22 - 2.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.67669294089816340444991237436, −9.540722386489089688130450440403, −9.061587878248231062650086661816, −8.550673237812053331625218464142, −7.45953058072387519896939068599, −6.67681124378915111632865818464, −5.40299195635250325180133665830, −4.24157865074013220076083476021, −1.91717467618529791570718850812, −1.49648140065739883071131887684, 0.921047934369314268279092625368, 2.26213108988716893335923718699, 3.54235736488196799909388692288, 5.52328502774778261315098256172, 6.49268983786385116625658414253, 7.33147724957678305711010311335, 8.184301056186100159565098923471, 8.885825369837843585298514884696, 9.797837737454326813811649842710, 10.65479576144465108490709565608

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