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Properties

Label 2-567-63.47-c1-0-15
Degree $2$
Conductor $567$
Sign $0.916 - 0.400i$
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  + (−1.22 + 0.707i)2-s + 2.44·5-s + (2.5 + 0.866i)7-s − 2.82i·8-s + (−2.99 + 1.73i)10-s − 1.41i·11-s + (4.5 − 2.59i)13-s + (−3.67 + 0.707i)14-s + (2.00 + 3.46i)16-s + (−2.44 − 4.24i)17-s + (1.5 + 0.866i)19-s + (1.00 + 1.73i)22-s − 5.65i·23-s + 0.999·25-s + (−3.67 + 6.36i)26-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + 1.09·5-s + (0.944 + 0.327i)7-s − 0.999i·8-s + (−0.948 + 0.547i)10-s − 0.426i·11-s + (1.24 − 0.720i)13-s + (−0.981 + 0.188i)14-s + (0.500 + 0.866i)16-s + (−0.594 − 1.02i)17-s + (0.344 + 0.198i)19-s + (0.213 + 0.369i)22-s − 1.17i·23-s + 0.199·25-s + (−0.720 + 1.24i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.916 - 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18225 + 0.247320i\)
\(L(\frac12)\) \(\approx\) \(1.18225 + 0.247320i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good2 \( 1 + (1.22 - 0.707i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 2.44T + 5T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.44 + 4.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.65iT - 23T^{2} \)
29 \( 1 + (2.44 + 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.67 - 6.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.12 - 10.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.44 + 1.41i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.44 - 4.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 + 1.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.07iT - 71T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.67 - 6.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.44 + 4.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9 + 5.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.65011039261290853620363084469, −9.680397572487884915090470260865, −8.926820183133272771381606303303, −8.312473170055198936382690081800, −7.43363979037443924273980427361, −6.26631296290212038167848212272, −5.57437200410905588881297469539, −4.24719456645803767679318495207, −2.69063004478973744775882442865, −1.12378064174167236051209178033, 1.45745426210340890765996525544, 2.03010064006059237676133456416, 3.95650409302015532512979972719, 5.23656119156737199851146161712, 6.01629935633614839713288584799, 7.27843878491923342367940872344, 8.397459193585075888091697183934, 9.059020768809128982142763855696, 9.775315978320175487910620638199, 10.73848585027735611682169432544

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