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Properties

Label 2-567-63.58-c1-0-6
Degree $2$
Conductor $567$
Sign $0.962 - 0.270i$
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  − 2.69·2-s + 5.28·4-s + (−0.794 − 1.37i)5-s + (−2.64 + 0.0963i)7-s − 8.87·8-s + (2.14 + 3.71i)10-s + (0.150 − 0.260i)11-s + (−1.40 + 2.43i)13-s + (7.13 − 0.260i)14-s + 13.3·16-s + (2.93 + 5.08i)17-s + (1.14 − 1.98i)19-s + (−4.19 − 7.27i)20-s + (−0.405 + 0.702i)22-s + (0.944 + 1.63i)23-s + ⋯
L(s)  = 1  − 1.90·2-s + 2.64·4-s + (−0.355 − 0.615i)5-s + (−0.999 + 0.0364i)7-s − 3.13·8-s + (0.677 + 1.17i)10-s + (0.0452 − 0.0784i)11-s + (−0.389 + 0.675i)13-s + (1.90 − 0.0695i)14-s + 3.34·16-s + (0.712 + 1.23i)17-s + (0.262 − 0.454i)19-s + (−0.939 − 1.62i)20-s + (−0.0864 + 0.149i)22-s + (0.196 + 0.341i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.962 - 0.270i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.962 - 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.448526 + 0.0619121i\)
\(L(\frac12)\) \(\approx\) \(0.448526 + 0.0619121i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0963i)T \)
good2 \( 1 + 2.69T + 2T^{2} \)
5 \( 1 + (0.794 + 1.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.150 + 0.260i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.40 - 2.43i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.93 - 5.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.14 + 1.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.944 - 1.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.26 - 2.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.81T + 31T^{2} \)
37 \( 1 + (-2.23 + 3.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.45 + 7.71i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.54 - 7.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.21T + 47T^{2} \)
53 \( 1 + (-1.00 - 1.74i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.88T + 59T^{2} \)
61 \( 1 - 7.57T + 61T^{2} \)
67 \( 1 - 0.712T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + (-5.83 - 10.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 1.66T + 79T^{2} \)
83 \( 1 + (2.71 + 4.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.67 - 8.10i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.28 - 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.54087204879815431850796965915, −9.650527304950468459205014067613, −9.116259487277813973550415051310, −8.344051803623816195688417575135, −7.43505786179578063357099553299, −6.66568093694992209653047731436, −5.66153130214581415464833478639, −3.77784846303759789923358263789, −2.39551526956320008215740957342, −0.894053063294227549349329949713, 0.66737402094787153529529167900, 2.55505911822643246321723439498, 3.36379724793488595934464380471, 5.60272252631564518666869170724, 6.71248380500341696692420259061, 7.32448210505210450737047499094, 8.020086784141918405631677016669, 9.163452037901872439323492675565, 9.773750624035691094956545446064, 10.39541435078228570434928000791

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