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Properties

Label 2-567-21.20-c3-0-79
Degree $2$
Conductor $567$
Sign $-0.808 - 0.588i$
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  − 2.67i·2-s + 0.835·4-s − 0.446·5-s + (−10.9 + 14.9i)7-s − 23.6i·8-s + 1.19i·10-s + 39.5i·11-s − 79.0i·13-s + (40.0 + 29.1i)14-s − 56.6·16-s + 9.74·17-s − 73.1i·19-s − 0.373·20-s + 105.·22-s + 146. i·23-s + ⋯
L(s)  = 1  − 0.946i·2-s + 0.104·4-s − 0.0399·5-s + (−0.588 + 0.808i)7-s − 1.04i·8-s + 0.0377i·10-s + 1.08i·11-s − 1.68i·13-s + (0.764 + 0.557i)14-s − 0.884·16-s + 0.139·17-s − 0.882i·19-s − 0.00417·20-s + 1.02·22-s + 1.32i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.808 - 0.588i$
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: \((2,\ 567,\ (\ :3/2),\ -0.808 - 0.588i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5536677303\)
\(L(\frac12)\) \(\approx\) \(0.5536677303\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.9 - 14.9i)T \)
good2 \( 1 + 2.67iT - 8T^{2} \)
5 \( 1 + 0.446T + 125T^{2} \)
11 \( 1 - 39.5iT - 1.33e3T^{2} \)
13 \( 1 + 79.0iT - 2.19e3T^{2} \)
17 \( 1 - 9.74T + 4.91e3T^{2} \)
19 \( 1 + 73.1iT - 6.85e3T^{2} \)
23 \( 1 - 146. iT - 1.21e4T^{2} \)
29 \( 1 + 154. iT - 2.43e4T^{2} \)
31 \( 1 + 11.4iT - 2.97e4T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + 106.T + 6.89e4T^{2} \)
43 \( 1 - 91.2T + 7.95e4T^{2} \)
47 \( 1 + 553.T + 1.03e5T^{2} \)
53 \( 1 - 239. iT - 1.48e5T^{2} \)
59 \( 1 + 252.T + 2.05e5T^{2} \)
61 \( 1 + 395. iT - 2.26e5T^{2} \)
67 \( 1 + 27.0T + 3.00e5T^{2} \)
71 \( 1 - 348. iT - 3.57e5T^{2} \)
73 \( 1 + 923. iT - 3.89e5T^{2} \)
79 \( 1 + 560.T + 4.93e5T^{2} \)
83 \( 1 - 562.T + 5.71e5T^{2} \)
89 \( 1 + 644.T + 7.04e5T^{2} \)
97 \( 1 - 493. iT - 9.12e5T^{2} \)
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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

−9.858111011517296223517219539869, −9.393005992907813067066030620681, −8.036085181952429967693575332093, −7.14645669664159723019958433411, −6.07616859830132479800915869229, −5.07726499722108595774709560266, −3.61909353379522776801549215768, −2.79797732200808753631660286840, −1.75446954611278501064347680842, −0.14958757973849594478396781386, 1.70843350116164865557132156626, 3.27644228246821149568193248009, 4.37722875415186879288811229301, 5.65014750008863076288954799094, 6.56337252014233344276594813543, 7.00734407629775646303481885859, 8.140529629638473336440131248768, 8.827751188094823668636668463479, 9.961201426294091019627005362581, 10.87695562780207782667672423622

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