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Properties

Label 2-567-63.16-c1-0-26
Degree $2$
Conductor $567$
Sign $-0.296 + 0.954i$
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 − 1.73i)4-s + (−2 − 1.73i)7-s + (−1 − 1.73i)13-s + (−1.99 − 3.46i)16-s + (3.5 − 6.06i)19-s − 5·25-s + (−5 + 1.73i)28-s + (−5.5 + 9.52i)31-s + (5 − 8.66i)37-s + (6.5 − 11.2i)43-s + (1.00 + 6.92i)49-s − 3.99·52-s + (6.5 + 11.2i)61-s − 7.99·64-s + (8 − 13.8i)67-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (−0.755 − 0.654i)7-s + (−0.277 − 0.480i)13-s + (−0.499 − 0.866i)16-s + (0.802 − 1.39i)19-s − 25-s + (−0.944 + 0.327i)28-s + (−0.987 + 1.71i)31-s + (0.821 − 1.42i)37-s + (0.991 − 1.71i)43-s + (0.142 + 0.989i)49-s − 0.554·52-s + (0.832 + 1.44i)61-s − 0.999·64-s + (0.977 − 1.69i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.296 + 0.954i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.296 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.757424 - 1.02851i\)
\(L(\frac12)\) \(\approx\) \(0.757424 - 1.02851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 1.73i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.5 - 9.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.5 + 11.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8 + 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.53384890986988536731388267521, −9.687850839425261836008690189877, −9.007718827836587567958558466615, −7.44097026221446841493736443232, −6.98324181287059409544472946414, −5.88012408238237531953535425725, −5.04396931776772678376488569330, −3.64447261449734748074486177092, −2.39363743426785320378985176239, −0.70416760303996808035751456476, 2.08312097283496730646427202482, 3.20271871530117632268876883518, 4.15697473406131112685907163469, 5.71772194824865225680159307629, 6.44893484596486964282560594211, 7.56466893603838790489181002051, 8.179289221383141528166678456506, 9.385546361136600896424394363798, 9.911788745813873248605678885613, 11.33754783024801363119247640029

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