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Properties

Label 2-567-63.47-c1-0-20
Degree $2$
Conductor $567$
Sign $-0.110 + 0.993i$
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 + (−0.5 − 2.59i)7-s + 2.23i·8-s + 4.47i·11-s + (1.5 − 0.866i)13-s + (3.87 + 4.47i)14-s + (0.499 + 0.866i)16-s + (−3.87 − 6.70i)17-s + (−3 − 1.73i)19-s + (−5.00 − 8.66i)22-s + 4.47i·23-s − 5·25-s + (−1.93 + 3.35i)26-s + (−7.50 − 2.59i)28-s + (−3.87 − 2.23i)29-s + ⋯
L(s)  = 1  + (−1.36 + 0.790i)2-s + (0.750 − 1.29i)4-s + (−0.188 − 0.981i)7-s + 0.790i·8-s + 1.34i·11-s + (0.416 − 0.240i)13-s + (1.03 + 1.19i)14-s + (0.124 + 0.216i)16-s + (−0.939 − 1.62i)17-s + (−0.688 − 0.397i)19-s + (−1.06 − 1.84i)22-s + 0.932i·23-s − 25-s + (−0.379 + 0.657i)26-s + (−1.41 − 0.490i)28-s + (−0.719 − 0.415i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.110 + 0.993i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.110 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.184991 - 0.206788i\)
\(L(\frac12)\) \(\approx\) \(0.184991 - 0.206788i\)
\(L(\frac{3}{2})\) 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 \( 1 + 5T^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + (-1.5 + 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.87 + 6.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.47iT - 23T^{2} \)
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 \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.87 + 6.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.87 + 6.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.87 + 2.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.87 + 6.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.5 - 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.94iT - 71T^{2} \)
73 \( 1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.87 + 6.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.74 - 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.15795976070159096451385414693, −9.569337214928191117374909817411, −8.807675230733221551500503365860, −7.66150571889870863861871909153, −7.16780385674940629381724572148, −6.47864158540082290693316065992, −5.06915720288724638683226214699, −3.85409000403147554641730157456, −1.92967121601403283118397270635, −0.24168670356783388117627951231, 1.64295505736839098282765837996, 2.73529916463730260517209129607, 3.96735536847710157080667329209, 5.75443296938047430036846845553, 6.45329649309305982106050566217, 8.067398350661893076803217852999, 8.520430708016927721547501788516, 9.115302731029622211328506240101, 10.11458814633247437438334769852, 11.00254815778219467833141389389

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