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Properties

Label 2-567-63.38-c1-0-14
Degree $2$
Conductor $567$
Sign $0.965 + 0.261i$
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  − 0.656i·2-s + 1.56·4-s + (−1.65 + 2.86i)5-s + (1.78 − 1.95i)7-s − 2.34i·8-s + (1.88 + 1.08i)10-s + (2.02 − 1.17i)11-s + (−1.36 + 0.790i)13-s + (−1.28 − 1.17i)14-s + 1.59·16-s + (−0.568 + 0.984i)17-s + (3.85 − 2.22i)19-s + (−2.59 + 4.5i)20-s + (−0.769 − 1.33i)22-s + (8.13 + 4.69i)23-s + ⋯
L(s)  = 1  − 0.464i·2-s + 0.784·4-s + (−0.740 + 1.28i)5-s + (0.674 − 0.738i)7-s − 0.828i·8-s + (0.595 + 0.343i)10-s + (0.611 − 0.353i)11-s + (−0.379 + 0.219i)13-s + (−0.342 − 0.313i)14-s + 0.399·16-s + (−0.137 + 0.238i)17-s + (0.884 − 0.510i)19-s + (−0.580 + 1.00i)20-s + (−0.164 − 0.284i)22-s + (1.69 + 0.979i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.965 + 0.261i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.965 + 0.261i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75331 - 0.233149i\)
\(L(\frac12)\) \(\approx\) \(1.75331 - 0.233149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.78 + 1.95i)T \)
good2 \( 1 + 0.656iT - 2T^{2} \)
5 \( 1 + (1.65 - 2.86i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.02 + 1.17i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.36 - 0.790i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.568 - 0.984i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.85 + 2.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-8.13 - 4.69i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.16 - 1.82i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.31iT - 31T^{2} \)
37 \( 1 + (-2.58 - 4.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.82 + 8.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.08 + 1.87i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.58T + 47T^{2} \)
53 \( 1 + (8.70 + 5.02i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.17T + 59T^{2} \)
61 \( 1 - 4.00iT - 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + 6.39iT - 71T^{2} \)
73 \( 1 + (9.25 + 5.34i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 1.23T + 79T^{2} \)
83 \( 1 + (-0.518 + 0.898i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.73 + 6.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.7 - 6.77i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.78362745613235493711821362587, −10.30754384301097786259693226723, −9.042628490659678614110550426011, −7.71520408250621767792961133749, −7.08787008643404434204894735716, −6.58284606968933415315904653220, −4.96173009440807276912549086392, −3.57943658055821541884673707477, −3.00421069466786977787219453523, −1.38090385509708288300992902466, 1.32602118171480339976357806503, 2.80522787414609620158412769819, 4.49160515879822341458702123543, 5.15389802482917070114115634083, 6.20613173912304348858639042554, 7.38153714212256876550118448959, 8.069636035167294882862081452940, 8.788185621044378678028645011302, 9.695514705868198993391385272581, 11.12865017344351872178085622334

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