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Properties

Label 2-567-9.7-c1-0-5
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  + (−1.32 + 2.29i)2-s + (−2.5 − 4.33i)4-s + (1.32 + 2.29i)5-s + (0.5 − 0.866i)7-s + 7.93·8-s − 7·10-s + (1.32 − 2.29i)11-s + (1 + 1.73i)13-s + (1.32 + 2.29i)14-s + (−5.49 + 9.52i)16-s + 7·19-s + (6.61 − 11.4i)20-s + (3.5 + 6.06i)22-s + (3.96 + 6.87i)23-s + (−1 + 1.73i)25-s − 5.29·26-s + ⋯
L(s)  = 1  + (−0.935 + 1.62i)2-s + (−1.25 − 2.16i)4-s + (0.591 + 1.02i)5-s + (0.188 − 0.327i)7-s + 2.80·8-s − 2.21·10-s + (0.398 − 0.690i)11-s + (0.277 + 0.480i)13-s + (0.353 + 0.612i)14-s + (−1.37 + 2.38i)16-s + 1.60·19-s + (1.47 − 2.56i)20-s + (0.746 + 1.29i)22-s + (0.827 + 1.43i)23-s + (−0.200 + 0.346i)25-s − 1.03·26-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: $\chi_{567} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.336945 + 0.925749i\)
\(L(\frac12)\) \(\approx\) \(0.336945 + 0.925749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 2.29i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.32 - 2.29i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.32 + 2.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (-3.96 - 6.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.64 - 4.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 + (1.32 + 2.29i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.93T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.93 - 13.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 18.5T + 89T^{2} \)
97 \( 1 + (-6 + 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.83280917659457455559598685099, −9.764381609410089934901888233530, −9.335808294571976032115683915082, −8.302624355901766592492031115991, −7.29406964875944349046920551781, −6.84347978611362862299762395545, −5.88626963209246009090443632817, −5.14753551535851252444716110218, −3.44361010995422362270805818530, −1.34722206450387431720409302345, 0.940417049047063968904108612557, 1.97695338729716011605404330780, 3.20829445335908572224602172141, 4.49384331146462366318721638305, 5.42585368077153635625500422181, 7.17430625861009816682813996908, 8.322130705159685853565238178871, 8.915006541109687133947429287191, 9.620626531185860850837736262684, 10.25005659911440755842250918357

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