Properties

Label 2-567-21.20-c3-0-60
Degree $2$
Conductor $567$
Sign $0.264 + 0.964i$
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  − 1.27i·2-s + 6.37·4-s − 3.19·5-s + (17.8 − 4.90i)7-s − 18.3i·8-s + 4.07i·10-s − 43.9i·11-s + 75.7i·13-s + (−6.26 − 22.7i)14-s + 27.5·16-s + 104.·17-s − 74.7i·19-s − 20.3·20-s − 56.1·22-s − 53.8i·23-s + ⋯
L(s)  = 1  − 0.451i·2-s + 0.796·4-s − 0.285·5-s + (0.964 − 0.264i)7-s − 0.810i·8-s + 0.128i·10-s − 1.20i·11-s + 1.61i·13-s + (−0.119 − 0.435i)14-s + 0.430·16-s + 1.49·17-s − 0.902i·19-s − 0.227·20-s − 0.544·22-s − 0.488i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.264 + 0.964i$
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: \((2,\ 567,\ (\ :3/2),\ 0.264 + 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.734809926\)
\(L(\frac12)\) \(\approx\) \(2.734809926\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.8 + 4.90i)T \)
good2 \( 1 + 1.27iT - 8T^{2} \)
5 \( 1 + 3.19T + 125T^{2} \)
11 \( 1 + 43.9iT - 1.33e3T^{2} \)
13 \( 1 - 75.7iT - 2.19e3T^{2} \)
17 \( 1 - 104.T + 4.91e3T^{2} \)
19 \( 1 + 74.7iT - 6.85e3T^{2} \)
23 \( 1 + 53.8iT - 1.21e4T^{2} \)
29 \( 1 + 30.4iT - 2.43e4T^{2} \)
31 \( 1 - 128. iT - 2.97e4T^{2} \)
37 \( 1 - 46.1T + 5.06e4T^{2} \)
41 \( 1 + 426.T + 6.89e4T^{2} \)
43 \( 1 - 332.T + 7.95e4T^{2} \)
47 \( 1 - 340.T + 1.03e5T^{2} \)
53 \( 1 - 235. iT - 1.48e5T^{2} \)
59 \( 1 - 544.T + 2.05e5T^{2} \)
61 \( 1 + 371. iT - 2.26e5T^{2} \)
67 \( 1 - 106.T + 3.00e5T^{2} \)
71 \( 1 + 974. iT - 3.57e5T^{2} \)
73 \( 1 + 576. iT - 3.89e5T^{2} \)
79 \( 1 - 21.3T + 4.93e5T^{2} \)
83 \( 1 - 337.T + 5.71e5T^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 - 73.3iT - 9.12e5T^{2} \)
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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

−10.45646391877832203455357024024, −9.342489639442137347900348225040, −8.338234408735248075607214597315, −7.47696125884791385237026212674, −6.61732623179530162939728677540, −5.55060856195017648756251003444, −4.27251439166110793648030206613, −3.28054351755478613332098432686, −1.99224135025807321797305602334, −0.886828064129646753008904304494, 1.33896930069929642112339416847, 2.51024717844231309683524986165, 3.83497539294204158590743148862, 5.35037292397082546122778646240, 5.70660427176299618943353249839, 7.20834685607928663987091893462, 7.79556641410438644396108398496, 8.289589770891375474748755554726, 9.882914783245519170784951507379, 10.42486863786061841285423923438

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