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Properties

Label 2-567-189.20-c1-0-2
Degree $2$
Conductor $567$
Sign $0.949 - 0.312i$
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  + (−0.471 − 1.29i)2-s + (0.0766 − 0.0643i)4-s + (0.459 + 2.60i)5-s + (−1.47 + 2.19i)7-s + (−2.50 − 1.44i)8-s + (3.15 − 1.82i)10-s + (0.681 + 0.120i)11-s + (0.171 − 0.472i)13-s + (3.54 + 0.869i)14-s + (−0.658 + 3.73i)16-s + (2.42 + 4.19i)17-s + (−1.03 − 0.597i)19-s + (0.202 + 0.170i)20-s + (−0.165 − 0.939i)22-s + (4.74 + 5.65i)23-s + ⋯
L(s)  = 1  + (−0.333 − 0.915i)2-s + (0.0383 − 0.0321i)4-s + (0.205 + 1.16i)5-s + (−0.556 + 0.831i)7-s + (−0.886 − 0.511i)8-s + (0.998 − 0.576i)10-s + (0.205 + 0.0362i)11-s + (0.0476 − 0.130i)13-s + (0.946 + 0.232i)14-s + (−0.164 + 0.933i)16-s + (0.587 + 1.01i)17-s + (−0.237 − 0.137i)19-s + (0.0453 + 0.0380i)20-s + (−0.0353 − 0.200i)22-s + (0.990 + 1.18i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.949 - 0.312i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.949 - 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14369 + 0.183575i\)
\(L(\frac12)\) \(\approx\) \(1.14369 + 0.183575i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.47 - 2.19i)T \)
good2 \( 1 + (0.471 + 1.29i)T + (-1.53 + 1.28i)T^{2} \)
5 \( 1 + (-0.459 - 2.60i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (-0.681 - 0.120i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (-0.171 + 0.472i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.42 - 4.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.03 + 0.597i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.74 - 5.65i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.19 - 6.03i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-5.17 - 6.16i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (3.71 + 6.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.25 + 1.18i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.81 - 10.2i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.93 - 2.45i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + (1.13 + 6.41i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-2.36 + 2.81i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (3.24 + 1.17i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (0.286 - 0.165i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.13 - 1.81i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.71 + 3.53i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (12.8 - 4.69i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-4.84 + 8.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.19 + 0.563i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.75697398979907450353699494187, −10.09165034610572026975133324440, −9.340527368792353788570938548475, −8.425577296517605728706147191820, −6.94716131753931408541288795096, −6.38535459422415635500134974382, −5.38511862460410284470244597014, −3.42791706252996699094225457408, −2.91810486527921187170044673832, −1.64223453434055037304433458889, 0.75682802109033268313670598959, 2.79147520869428659858466315849, 4.26540542781327772117159925497, 5.28063691586618896468447107870, 6.35696630132789570366505729721, 7.08722738342044898413081697900, 8.039005076855623436483261976217, 8.805497596543486899694217532743, 9.549269330895040999858586968797, 10.48217009054840939622224903367

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