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Properties

Label 2-567-189.185-c1-0-3
Degree $2$
Conductor $567$
Sign $-0.198 - 0.980i$
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  + (1.57 − 0.277i)2-s + (0.513 − 0.186i)4-s + (−1.78 + 0.649i)5-s + (−1.28 + 2.31i)7-s + (−2.00 + 1.15i)8-s + (−2.62 + 1.51i)10-s + (−0.432 + 1.18i)11-s + (0.326 + 0.898i)13-s + (−1.37 + 3.99i)14-s + (−3.67 + 3.08i)16-s + (2.00 + 3.47i)17-s + (6.64 + 3.83i)19-s + (−0.794 + 0.666i)20-s + (−0.350 + 1.98i)22-s + (−5.63 − 0.994i)23-s + ⋯
L(s)  = 1  + (1.11 − 0.195i)2-s + (0.256 − 0.0934i)4-s + (−0.797 + 0.290i)5-s + (−0.485 + 0.874i)7-s + (−0.710 + 0.410i)8-s + (−0.829 + 0.478i)10-s + (−0.130 + 0.357i)11-s + (0.0906 + 0.249i)13-s + (−0.368 + 1.06i)14-s + (−0.918 + 0.770i)16-s + (0.486 + 0.843i)17-s + (1.52 + 0.880i)19-s + (−0.177 + 0.149i)20-s + (−0.0746 + 0.423i)22-s + (−1.17 − 0.207i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.918703 + 1.12367i\)
\(L(\frac12)\) \(\approx\) \(0.918703 + 1.12367i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 - 2.31i)T \)
good2 \( 1 + (-1.57 + 0.277i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (1.78 - 0.649i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (0.432 - 1.18i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.326 - 0.898i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.00 - 3.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.64 - 3.83i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.63 + 0.994i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.22 + 3.37i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.04 + 8.36i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + 7.99T + 37T^{2} \)
41 \( 1 + (-3.01 + 1.09i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.111 - 0.632i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-8.03 - 2.92i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-11.3 - 6.54i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.65 - 1.38i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.05 - 2.89i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.371 + 2.10i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.154 - 0.0893i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.5iT - 73T^{2} \)
79 \( 1 + (0.211 + 1.20i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-5.11 - 1.86i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (2.81 - 4.88i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.97 + 1.05i)T + (91.1 - 33.1i)T^{2} \)
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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

−11.44530874795444069232533641837, −10.18038923768672619774138105607, −9.292076552176903182109350956381, −8.245723147565739492042299075914, −7.39868025898588156168220304183, −5.99092976858463785848117395684, −5.54327351270129632024621355445, −4.10487830490078657902345323599, −3.54127079278111262701403427956, −2.31754712934591280172789371249, 0.57595106660466399974539594060, 3.16641407649884354104876829448, 3.75863902407588158354402391668, 4.86026921557358035537838652372, 5.60617930908900381830830480150, 6.89053065475735992151960870392, 7.50007379320645703116908505900, 8.712261554825097573571201503658, 9.662264906318454902400760294005, 10.58435480876017256711387010561

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