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Properties

Label 2-567-7.4-c1-0-17
Degree $2$
Conductor $567$
Sign $-0.979 - 0.202i$
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  + (−1.29 − 2.24i)2-s + (−2.34 + 4.06i)4-s + (−1.14 − 1.97i)5-s + (2.45 − 0.989i)7-s + 6.97·8-s + (−2.95 + 5.11i)10-s + (1.47 − 2.56i)11-s + 4.26·13-s + (−5.39 − 4.21i)14-s + (−4.32 − 7.49i)16-s + (0.764 − 1.32i)17-s + (−3.69 − 6.39i)19-s + 10.7·20-s − 7.64·22-s + (3.07 + 5.32i)23-s + ⋯
L(s)  = 1  + (−0.914 − 1.58i)2-s + (−1.17 + 2.03i)4-s + (−0.510 − 0.884i)5-s + (0.927 − 0.374i)7-s + 2.46·8-s + (−0.934 + 1.61i)10-s + (0.445 − 0.771i)11-s + 1.18·13-s + (−1.44 − 1.12i)14-s + (−1.08 − 1.87i)16-s + (0.185 − 0.321i)17-s + (−0.846 − 1.46i)19-s + 2.39·20-s − 1.63·22-s + (0.641 + 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.979 - 0.202i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.979 - 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0811394 + 0.794087i\)
\(L(\frac12)\) \(\approx\) \(0.0811394 + 0.794087i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 + 0.989i)T \)
good2 \( 1 + (1.29 + 2.24i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.14 + 1.97i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.47 + 2.56i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.26T + 13T^{2} \)
17 \( 1 + (-0.764 + 1.32i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.69 + 6.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.07 - 5.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.34T + 29T^{2} \)
31 \( 1 + (3.11 - 5.38i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.58 + 6.21i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.89T + 41T^{2} \)
43 \( 1 - 0.834T + 43T^{2} \)
47 \( 1 + (2.91 + 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.71 - 6.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.31 + 4.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.56 - 6.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.66 + 2.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.160T + 71T^{2} \)
73 \( 1 + (-0.190 + 0.329i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.97 - 6.88i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.29T + 83T^{2} \)
89 \( 1 + (-3.02 - 5.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.32T + 97T^{2} \)
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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

−10.60326633351614945220489364695, −9.247690744055262978854226996565, −8.713739201457723557986293555397, −8.230969955003856152059665021997, −7.07350210043332575035330976142, −5.20442330910704329902035238741, −4.15690507806590389127221330874, −3.32299023482943864578219838519, −1.67962559926462854111824903477, −0.69491116027861631629233463245, 1.64686653680827616898102988237, 3.85437564566033959001911822084, 5.04377372336568548520017354712, 6.17763829320418665201814393965, 6.74796466564479736261787165812, 7.78905575898734090438604795743, 8.289772088724781405544398818394, 9.094326454606083495736008949070, 10.23152877811182990524707837597, 10.84331966763799627996189629843

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