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Properties

Label 2-567-63.41-c1-0-27
Degree $2$
Conductor $567$
Sign $-0.0155 + 0.999i$
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  + (1.93 − 1.11i)2-s + (1.5 − 2.59i)4-s + (1.93 − 3.35i)5-s + (−0.5 + 2.59i)7-s − 2.23i·8-s − 8.66i·10-s + (1.93 − 1.11i)11-s + (−3 − 1.73i)13-s + (1.93 + 5.59i)14-s + (0.499 + 0.866i)16-s + 5.19i·19-s + (−5.80 − 10.0i)20-s + (2.5 − 4.33i)22-s + (−1.93 − 1.11i)23-s + (−5.00 − 8.66i)25-s − 7.74·26-s + ⋯
L(s)  = 1  + (1.36 − 0.790i)2-s + (0.750 − 1.29i)4-s + (0.866 − 1.50i)5-s + (−0.188 + 0.981i)7-s − 0.790i·8-s − 2.73i·10-s + (0.583 − 0.337i)11-s + (−0.832 − 0.480i)13-s + (0.517 + 1.49i)14-s + (0.124 + 0.216i)16-s + 1.19i·19-s + (−1.29 − 2.25i)20-s + (0.533 − 0.923i)22-s + (−0.403 − 0.233i)23-s + (−1.00 − 1.73i)25-s − 1.51·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0155 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0155 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.0155 + 0.999i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.0155 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31509 - 2.35148i\)
\(L(\frac12)\) \(\approx\) \(2.31509 - 2.35148i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
good2 \( 1 + (-1.93 + 1.11i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.93 + 3.35i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.93 + 1.11i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 + (1.93 + 1.11i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.87 - 2.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-1.93 + 3.35i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.87 - 6.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.94iT - 53T^{2} \)
59 \( 1 + (3.87 - 6.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.87 - 6.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (12 - 6.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.68744801785195940720432982234, −9.702197092766448208605764273452, −8.999148366587893252802226064900, −8.053652801135659457583765113585, −6.19502444382945117239906528656, −5.58945848832521642738595927511, −4.95943732823055040046122508188, −3.88553356712731520371240458308, −2.53909191092273648893652756694, −1.51752136278362840118567896613, 2.34281753852418537369298031375, 3.46694001175117970725885928889, 4.40168858837561560161912127880, 5.52267600196777202611959129280, 6.68700996823244139425876799979, 6.80963969760706032240848351285, 7.66008300448257133868180718382, 9.507520498707594459640829773422, 10.06376523929886579078037121190, 11.12163441752151504714996034716

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