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Properties

Label 2-567-63.4-c1-0-2
Degree $2$
Conductor $567$
Sign $-0.997 + 0.0746i$
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  + (−1.34 + 2.33i)2-s + (−2.64 − 4.57i)4-s − 1.58·5-s + (1.40 − 2.24i)7-s + 8.87·8-s + (2.14 − 3.71i)10-s + 0.300·11-s + (−1.40 + 2.43i)13-s + (3.34 + 6.31i)14-s + (−6.69 + 11.5i)16-s + (−2.93 + 5.08i)17-s + (1.14 + 1.98i)19-s + (4.19 + 7.27i)20-s + (−0.405 + 0.702i)22-s + 1.88·23-s + ⋯
L(s)  = 1  + (−0.954 + 1.65i)2-s + (−1.32 − 2.28i)4-s − 0.710·5-s + (0.531 − 0.847i)7-s + 3.13·8-s + (0.677 − 1.17i)10-s + 0.0905·11-s + (−0.389 + 0.675i)13-s + (0.893 + 1.68i)14-s + (−1.67 + 2.89i)16-s + (−0.712 + 1.23i)17-s + (0.262 + 0.454i)19-s + (0.939 + 1.62i)20-s + (−0.0864 + 0.149i)22-s + 0.393·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.997 + 0.0746i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.997 + 0.0746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0178391 - 0.477207i\)
\(L(\frac12)\) \(\approx\) \(0.0178391 - 0.477207i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 2.24i)T \)
good2 \( 1 + (1.34 - 2.33i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 - 0.300T + 11T^{2} \)
13 \( 1 + (1.40 - 2.43i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.93 - 5.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.14 - 1.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.88T + 23T^{2} \)
29 \( 1 + (1.26 + 2.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.40 - 4.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.23 - 3.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.45 - 7.71i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.54 - 7.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.60 + 2.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.00 - 1.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.44 - 4.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.78 - 6.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.356 + 0.616i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + (-5.83 + 10.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.833 - 1.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.71 - 4.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.67 - 8.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.28 - 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.83250302735231362097188813459, −10.08719101545092253565711771500, −9.158745988978627207170770122764, −8.212645784448158930266373514354, −7.75160238793135063105888954226, −6.88060136728487921795250385424, −6.12220216089425062403629422269, −4.80799066992221110620881414263, −4.09638225068210606358676429669, −1.37230202456763845998654221197, 0.41064369656042553662034201201, 2.13908428662335863450845224829, 3.02596881204027081117364815903, 4.21916549435895103407059861326, 5.24888869374364071778697137155, 7.25473037162746469610594838148, 7.953990362110690817254011875756, 8.884644603977449393879922457204, 9.374993533465267171063637633889, 10.43649950868117047131073497990

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