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Properties

Label 2-567-63.58-c1-0-25
Degree $2$
Conductor $567$
Sign $0.260 + 0.965i$
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  + 1.46·2-s + 0.133·4-s + (−0.296 − 0.514i)5-s + (−0.0665 − 2.64i)7-s − 2.72·8-s + (−0.433 − 0.750i)10-s + (2.23 − 3.86i)11-s + (2.25 − 3.90i)13-s + (−0.0971 − 3.86i)14-s − 4.24·16-s + (−0.136 − 0.236i)17-s + (−1.43 + 2.48i)19-s + (−0.0394 − 0.0684i)20-s + (3.25 − 5.64i)22-s + (2.52 + 4.37i)23-s + ⋯
L(s)  = 1  + 1.03·2-s + 0.0665·4-s + (−0.132 − 0.229i)5-s + (−0.0251 − 0.999i)7-s − 0.964·8-s + (−0.137 − 0.237i)10-s + (0.672 − 1.16i)11-s + (0.626 − 1.08i)13-s + (−0.0259 − 1.03i)14-s − 1.06·16-s + (−0.0331 − 0.0574i)17-s + (−0.328 + 0.569i)19-s + (−0.00883 − 0.0152i)20-s + (0.694 − 1.20i)22-s + (0.526 + 0.912i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.260 + 0.965i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.260 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58784 - 1.21680i\)
\(L(\frac12)\) \(\approx\) \(1.58784 - 1.21680i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0665 + 2.64i)T \)
good2 \( 1 - 1.46T + 2T^{2} \)
5 \( 1 + (0.296 + 0.514i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.23 + 3.86i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.25 + 3.90i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.136 + 0.236i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.43 - 2.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.52 - 4.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.176 - 0.305i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.51T + 31T^{2} \)
37 \( 1 + (-3.32 + 5.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.44 - 9.43i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.69 + 2.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 + (-5.66 - 9.80i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.05T + 59T^{2} \)
61 \( 1 + 2.73T + 61T^{2} \)
67 \( 1 - 5.86T + 67T^{2} \)
71 \( 1 + 2.60T + 71T^{2} \)
73 \( 1 + (5.55 + 9.62i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + (-8.27 - 14.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.68 - 4.65i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.13 - 1.96i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.78154573645329445400163816751, −9.760060989250125495455908618745, −8.671126421728940346006244197981, −7.971465044139935109433025862076, −6.60329695489489407654866871464, −5.87049649925064202208754514355, −4.84030127523679185061860983234, −3.80223298498029479262976145162, −3.18909649567410138811963264065, −0.876359553905391236420701543855, 2.07568563717449980737874038988, 3.33612676183136691489729675805, 4.44019114723370315064225436348, 5.12243449147293868634570045290, 6.41459217917903152595739213416, 6.84502520383115124979639965478, 8.531176512081164112562929281047, 9.072333632574714676444342121308, 9.972106661065821893453365910729, 11.43301912431372770433082068987

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