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Properties

Label 2-567-7.4-c1-0-4
Degree $2$
Conductor $567$
Sign $-0.999 - 0.0109i$
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  + (1.19 + 2.06i)2-s + (−1.84 + 3.20i)4-s + (1.46 + 2.52i)5-s + (−2.21 + 1.44i)7-s − 4.05·8-s + (−3.48 + 6.03i)10-s + (0.676 − 1.17i)11-s + 1.46·13-s + (−5.62 − 2.86i)14-s + (−1.13 − 1.96i)16-s + (1.65 − 2.86i)17-s + (−1.10 − 1.91i)19-s − 10.7·20-s + 3.23·22-s + (−1.31 − 2.27i)23-s + ⋯
L(s)  = 1  + (0.843 + 1.46i)2-s + (−0.924 + 1.60i)4-s + (0.653 + 1.13i)5-s + (−0.838 + 0.544i)7-s − 1.43·8-s + (−1.10 + 1.90i)10-s + (0.204 − 0.353i)11-s + 0.406·13-s + (−1.50 − 0.766i)14-s + (−0.284 − 0.492i)16-s + (0.401 − 0.695i)17-s + (−0.253 − 0.438i)19-s − 2.41·20-s + 0.688·22-s + (−0.274 − 0.474i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.999 - 0.0109i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.999 - 0.0109i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0119533 + 2.17718i\)
\(L(\frac12)\) \(\approx\) \(0.0119533 + 2.17718i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 1.44i)T \)
good2 \( 1 + (-1.19 - 2.06i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.46 - 2.52i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.676 + 1.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.46T + 13T^{2} \)
17 \( 1 + (-1.65 + 2.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.10 + 1.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.31 + 2.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.04T + 29T^{2} \)
31 \( 1 + (1.63 - 2.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.43 - 9.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.80T + 41T^{2} \)
43 \( 1 - 4.34T + 43T^{2} \)
47 \( 1 + (1.98 + 3.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.22 - 5.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.10 + 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.279 + 0.484i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.40 - 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (-5.22 + 9.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.383 + 0.664i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.96T + 83T^{2} \)
89 \( 1 + (-3.20 - 5.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.28T + 97T^{2} \)
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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

−11.23305936755595681948098057886, −10.18262646835891850964201367950, −9.261719902874860800745865281092, −8.282798039276060104272840722996, −7.18272491306839089583732706529, −6.43675212187789768204593193209, −6.03910320584942157072475992184, −4.99326940685800285001975829473, −3.61292252974232714139977848643, −2.69401729284830347526489162868, 1.02204252587025229301811513436, 2.13919695237118429092156357502, 3.61859987527044358009739382547, 4.27211065725564304848998304800, 5.43565767656641859748005690290, 6.16324067412067999606746799261, 7.72010247542246667352020877739, 9.112796886842903172412904467134, 9.638360765618598998707362890478, 10.40487701623308798533027523497

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