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Properties

Label 2-567-9.4-c1-0-14
Degree $2$
Conductor $567$
Sign $0.766 - 0.642i$
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  + (0.866 + 1.5i)2-s + (−0.5 + 0.866i)4-s + (0.866 − 1.5i)5-s + (−0.5 − 0.866i)7-s + 1.73·8-s + 3·10-s + (−0.866 − 1.5i)11-s + (−1 + 1.73i)13-s + (0.866 − 1.5i)14-s + (2.49 + 4.33i)16-s + 6.92·17-s + 5·19-s + (0.866 + 1.5i)20-s + (1.5 − 2.59i)22-s + (0.866 − 1.5i)23-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s + (−0.250 + 0.433i)4-s + (0.387 − 0.670i)5-s + (−0.188 − 0.327i)7-s + 0.612·8-s + 0.948·10-s + (−0.261 − 0.452i)11-s + (−0.277 + 0.480i)13-s + (0.231 − 0.400i)14-s + (0.624 + 1.08i)16-s + 1.68·17-s + 1.14·19-s + (0.193 + 0.335i)20-s + (0.319 − 0.553i)22-s + (0.180 − 0.312i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15085 + 0.782846i\)
\(L(\frac12)\) \(\approx\) \(2.15085 + 0.782846i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.866 + 1.5i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (-0.866 + 1.5i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.19 + 9i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-2.59 + 4.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.46 - 6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 13.8T + 53T^{2} \)
59 \( 1 + (3.46 - 6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.19T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.19 + 9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.66T + 89T^{2} \)
97 \( 1 + (-2 - 3.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.76366001711602105992866056290, −9.851646908800106042201790445536, −9.022310648496691638979141784621, −7.77258847273088795250527543130, −7.31916807953055092475224895583, −6.03345830149060292034882977508, −5.47406321642738930123320060712, −4.58461464960830988142800607231, −3.33466127005194170598918501691, −1.37114617351508921125778695066, 1.62030256866595147010950196573, 2.92541520826229938449365536738, 3.46622106553540447445582025502, 5.00053325509720938335022166426, 5.71456967415541527279950685135, 7.18006444722374662995972729642, 7.76740824301625058812568832826, 9.311139167976071055689163871343, 10.08784914108900041477316716644, 10.66870614531820493126834219682

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