Properties

Label 2-567-189.131-c1-0-9
Degree $2$
Conductor $567$
Sign $-0.673 - 0.739i$
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  + (1.44 + 1.72i)2-s + (−0.536 + 3.03i)4-s + (2.61 + 2.19i)5-s + (−0.126 + 2.64i)7-s + (−2.12 + 1.22i)8-s + 7.69i·10-s + (−1.63 − 1.94i)11-s + (−2.11 − 5.80i)13-s + (−4.74 + 3.61i)14-s + (0.606 + 0.220i)16-s + 2.39·17-s − 1.24i·19-s + (−8.07 + 6.77i)20-s + (0.996 − 5.65i)22-s + (−0.769 − 2.11i)23-s + ⋯
L(s)  = 1  + (1.02 + 1.22i)2-s + (−0.268 + 1.51i)4-s + (1.16 + 0.981i)5-s + (−0.0479 + 0.998i)7-s + (−0.750 + 0.433i)8-s + 2.43i·10-s + (−0.493 − 0.587i)11-s + (−0.585 − 1.60i)13-s + (−1.26 + 0.965i)14-s + (0.151 + 0.0551i)16-s + 0.581·17-s − 0.285i·19-s + (−1.80 + 1.51i)20-s + (0.212 − 1.20i)22-s + (−0.160 − 0.440i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.673 - 0.739i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.673 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14731 + 2.59649i\)
\(L(\frac12)\) \(\approx\) \(1.14731 + 2.59649i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.126 - 2.64i)T \)
good2 \( 1 + (-1.44 - 1.72i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (-2.61 - 2.19i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (1.63 + 1.94i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (2.11 + 5.80i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 - 2.39T + 17T^{2} \)
19 \( 1 + 1.24iT - 19T^{2} \)
23 \( 1 + (0.769 + 2.11i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.685 - 1.88i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (1.55 + 0.273i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (5.01 + 8.68i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.39 + 1.96i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.110 + 0.625i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-2.09 - 11.8i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (4.15 - 2.39i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.59 - 0.943i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-7.26 + 1.28i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.14 + 5.15i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (4.45 + 2.57i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-11.6 - 6.70i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.740 + 0.621i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-2.38 - 0.867i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 - 9.95T + 89T^{2} \)
97 \( 1 + (-7.93 + 1.39i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.92995007237776184266512303097, −10.26243631704332029064052481533, −9.206009798817640989021389985296, −8.037270313035941273540760017546, −7.28606463495892970310484126011, −6.14639346233108279195820878140, −5.72399509463285834915135352211, −5.07418643638965934871429958372, −3.31152360921482785227228867298, −2.52214965050431831530891602604, 1.39556813284848367050939734595, 2.20747537498632308603370584376, 3.75223760405214435375802981056, 4.69930495142624680035172754146, 5.27194288204582980181896762426, 6.50009796852592676705774385821, 7.70632654690059900699022118717, 9.144059028836473346037918798579, 9.927375130529217233953670978728, 10.32008940679772191363673880238

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