Properties

Label 2-567-567.104-c1-0-11
Degree $2$
Conductor $567$
Sign $-0.649 + 0.760i$
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  + (−2.00 − 1.49i)2-s + (−1.71 − 0.269i)3-s + (1.22 + 4.07i)4-s + (−3.15 − 2.07i)5-s + (3.03 + 3.09i)6-s + (−0.572 + 2.58i)7-s + (1.92 − 5.29i)8-s + (2.85 + 0.921i)9-s + (3.22 + 8.86i)10-s + (−0.664 + 1.32i)11-s + (−0.990 − 7.30i)12-s + (−2.20 − 0.950i)13-s + (5.00 − 4.32i)14-s + (4.83 + 4.39i)15-s + (−4.67 + 3.07i)16-s + (4.99 + 4.19i)17-s + ⋯
L(s)  = 1  + (−1.41 − 1.05i)2-s + (−0.987 − 0.155i)3-s + (0.610 + 2.03i)4-s + (−1.40 − 0.927i)5-s + (1.23 + 1.26i)6-s + (−0.216 + 0.976i)7-s + (0.681 − 1.87i)8-s + (0.951 + 0.307i)9-s + (1.02 + 2.80i)10-s + (−0.200 + 0.399i)11-s + (−0.285 − 2.10i)12-s + (−0.610 − 0.263i)13-s + (1.33 − 1.15i)14-s + (1.24 + 1.13i)15-s + (−1.16 + 0.768i)16-s + (1.21 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.649 + 0.760i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0903535 - 0.195958i\)
\(L(\frac12)\) \(\approx\) \(0.0903535 - 0.195958i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.269i)T \)
7 \( 1 + (0.572 - 2.58i)T \)
good2 \( 1 + (2.00 + 1.49i)T + (0.573 + 1.91i)T^{2} \)
5 \( 1 + (3.15 + 2.07i)T + (1.98 + 4.59i)T^{2} \)
11 \( 1 + (0.664 - 1.32i)T + (-6.56 - 8.82i)T^{2} \)
13 \( 1 + (2.20 + 0.950i)T + (8.92 + 9.45i)T^{2} \)
17 \( 1 + (-4.99 - 4.19i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-1.18 - 1.41i)T + (-3.29 + 18.7i)T^{2} \)
23 \( 1 + (4.02 + 3.79i)T + (1.33 + 22.9i)T^{2} \)
29 \( 1 + (0.617 - 5.28i)T + (-28.2 - 6.68i)T^{2} \)
31 \( 1 + (-1.44 + 6.08i)T + (-27.7 - 13.9i)T^{2} \)
37 \( 1 + (0.361 - 2.05i)T + (-34.7 - 12.6i)T^{2} \)
41 \( 1 + (-2.35 - 3.15i)T + (-11.7 + 39.2i)T^{2} \)
43 \( 1 + (0.560 + 9.62i)T + (-42.7 + 4.99i)T^{2} \)
47 \( 1 + (-0.725 + 0.171i)T + (42.0 - 21.0i)T^{2} \)
53 \( 1 + (9.25 + 5.34i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.37 - 3.20i)T + (35.2 - 47.3i)T^{2} \)
61 \( 1 + (-8.54 - 2.55i)T + (50.9 + 33.5i)T^{2} \)
67 \( 1 + (8.70 - 1.01i)T + (65.1 - 15.4i)T^{2} \)
71 \( 1 + (1.67 + 4.61i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (1.43 - 3.95i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (-7.48 + 10.0i)T + (-22.6 - 75.6i)T^{2} \)
83 \( 1 + (-3.78 + 5.08i)T + (-23.8 - 79.5i)T^{2} \)
89 \( 1 + (2.75 + 1.00i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (0.709 + 1.07i)T + (-38.4 + 89.0i)T^{2} \)
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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

−10.37641617309941259409949110358, −9.757356024856667061751578512831, −8.674916948875938724099731544381, −7.989777826391946412681728122637, −7.37036277900977322909118080166, −5.80943294595717950771454357747, −4.61693634219953470792233441203, −3.40624765938746000983560949208, −1.77821073877370288266200725328, −0.36129652779759643369666350685, 0.77480310919779160854727642167, 3.44808156016529854045197459501, 4.76122847096017061207225702004, 6.03291809525129574388038878712, 6.91977969141122098659187773649, 7.53172632384025962005645792520, 7.896732615493299272396925505131, 9.504523002496719091501465634203, 10.07046296370333903806598606386, 10.88721809654059647995413562106

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