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Properties

Label 2-567-63.4-c1-0-18
Degree $2$
Conductor $567$
Sign $0.704 - 0.709i$
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  + (−0.764 + 1.32i)2-s + (−0.167 − 0.290i)4-s + 2.82·5-s + (0.955 − 2.46i)7-s − 2.54·8-s + (−2.15 + 3.73i)10-s + 3.63·11-s + (2.81 − 4.87i)13-s + (2.53 + 3.14i)14-s + (2.27 − 3.94i)16-s + (−1.60 + 2.77i)17-s + (−2.03 − 3.52i)19-s + (−0.473 − 0.820i)20-s + (−2.77 + 4.81i)22-s + 4.70·23-s + ⋯
L(s)  = 1  + (−0.540 + 0.935i)2-s + (−0.0838 − 0.145i)4-s + 1.26·5-s + (0.361 − 0.932i)7-s − 0.899·8-s + (−0.682 + 1.18i)10-s + 1.09·11-s + (0.780 − 1.35i)13-s + (0.677 + 0.841i)14-s + (0.569 − 0.986i)16-s + (−0.388 + 0.672i)17-s + (−0.466 − 0.808i)19-s + (−0.105 − 0.183i)20-s + (−0.592 + 1.02i)22-s + 0.980·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.704 - 0.709i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.704 - 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38625 + 0.576966i\)
\(L(\frac12)\) \(\approx\) \(1.38625 + 0.576966i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.955 + 2.46i)T \)
good2 \( 1 + (0.764 - 1.32i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
11 \( 1 - 3.63T + 11T^{2} \)
13 \( 1 + (-2.81 + 4.87i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.60 - 2.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.03 + 3.52i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.70T + 23T^{2} \)
29 \( 1 + (2.16 + 3.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.79 + 3.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.15 - 3.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.57 - 2.72i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.59 - 7.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.42 - 4.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.06 - 12.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.750 + 1.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.60 - 11.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.34 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.91T + 71T^{2} \)
73 \( 1 + (1.46 - 2.53i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.446 - 0.773i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.02 + 6.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.82 + 4.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.56 + 4.44i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.72122924456382738128083345817, −9.738072587596312325958166881706, −9.011463655004701079207682962653, −8.170352996781359739293253710488, −7.25506687203144889448505987917, −6.30141282627300114002806554874, −5.83405807021219144171615241542, −4.37164871560508115054751094666, −2.95610048658099125303340324439, −1.21787758154654761389105972818, 1.59721895427643320941095959631, 2.10628462665347109519396828305, 3.57244002050278653392051479601, 5.14429677965995218142823912827, 6.13780355822554693045993344191, 6.74793491739838375007343500773, 8.618994580049385506120526240371, 9.187044382764297005687217293206, 9.537034111351160214344238470500, 10.73719408127191226817579792794

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