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Properties

Label 2-567-63.25-c1-0-8
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: $\chi_{567} (298, \cdot )$
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

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

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