Properties

Label 2-567-63.38-c1-0-26
Degree $2$
Conductor $567$
Sign $-0.827 + 0.561i$
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  − 1.83i·2-s − 1.36·4-s + (1.90 − 3.29i)5-s + (0.317 + 2.62i)7-s − 1.16i·8-s + (−6.04 − 3.49i)10-s + (1.00 − 0.582i)11-s + (4.79 − 2.77i)13-s + (4.81 − 0.582i)14-s − 4.86·16-s + (−1.58 + 2.75i)17-s + (−0.546 + 0.315i)19-s + (−2.59 + 4.50i)20-s + (−1.06 − 1.85i)22-s + (−1.52 − 0.880i)23-s + ⋯
L(s)  = 1  − 1.29i·2-s − 0.682·4-s + (0.851 − 1.47i)5-s + (0.120 + 0.992i)7-s − 0.412i·8-s + (−1.91 − 1.10i)10-s + (0.304 − 0.175i)11-s + (1.33 − 0.768i)13-s + (1.28 − 0.155i)14-s − 1.21·16-s + (−0.385 + 0.667i)17-s + (−0.125 + 0.0724i)19-s + (−0.580 + 1.00i)20-s + (−0.227 − 0.394i)22-s + (−0.318 − 0.183i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.827 + 0.561i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.525045 - 1.70877i\)
\(L(\frac12)\) \(\approx\) \(0.525045 - 1.70877i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.317 - 2.62i)T \)
good2 \( 1 + 1.83iT - 2T^{2} \)
5 \( 1 + (-1.90 + 3.29i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.00 + 0.582i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.79 + 2.77i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.58 - 2.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.546 - 0.315i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.52 + 0.880i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.18 - 2.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.27iT - 31T^{2} \)
37 \( 1 + (2.11 + 3.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.28 + 3.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.61 - 6.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.54T + 47T^{2} \)
53 \( 1 + (0.0627 + 0.0362i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.98T + 59T^{2} \)
61 \( 1 - 8.08iT - 61T^{2} \)
67 \( 1 - 5.09T + 67T^{2} \)
71 \( 1 - 4.76iT - 71T^{2} \)
73 \( 1 + (-4.84 - 2.79i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 17.0T + 79T^{2} \)
83 \( 1 + (5.08 - 8.80i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.77 + 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.596 + 0.344i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.45570881369881824020881491647, −9.597005446968118131990770967459, −8.707142993731948537279908485744, −8.440932595754920047918884128312, −6.40891496886093690803873400269, −5.62634954383999393698423674843, −4.59673162482232458199916693462, −3.37400383133774296846382590047, −2.02437243152652984721226303184, −1.14245127268700299734725678936, 2.01190230457740070590023664335, 3.50836004016527483745364906612, 4.77968312600438141198947721390, 6.19166694141667797713774431218, 6.51898752514013505984229255086, 7.21217919635224747884834269493, 8.150909405913302743492227020675, 9.276056323431976429723745719986, 10.18256335476411675131507423446, 11.04503894087954112090159873986

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