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Properties

Label 2-567-9.2-c2-0-3
Degree $2$
Conductor $567$
Sign $-0.342 - 0.939i$
Analytic cond. $15.4496$
Root an. cond. $3.93060$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.03 − 1.75i)2-s + (4.14 + 7.18i)4-s + (−1.07 + 0.620i)5-s + (−1.32 + 2.29i)7-s − 15.0i·8-s + 4.35·10-s + (−6.07 − 3.50i)11-s + (5.82 + 10.0i)13-s + (8.03 − 4.63i)14-s + (−9.79 + 16.9i)16-s − 4.52i·17-s + 16.2·19-s + (−8.91 − 5.14i)20-s + (12.2 + 21.2i)22-s + (22.1 − 12.7i)23-s + ⋯
L(s)  = 1  + (−1.51 − 0.876i)2-s + (1.03 + 1.79i)4-s + (−0.215 + 0.124i)5-s + (−0.188 + 0.327i)7-s − 1.88i·8-s + 0.435·10-s + (−0.552 − 0.318i)11-s + (0.447 + 0.775i)13-s + (0.573 − 0.331i)14-s + (−0.611 + 1.05i)16-s − 0.266i·17-s + 0.854·19-s + (−0.445 − 0.257i)20-s + (0.558 + 0.967i)22-s + (0.962 − 0.555i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(15.4496\)
Root analytic conductor: \(3.93060\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2115024863\)
\(L(\frac12)\) \(\approx\) \(0.2115024863\)
\(L(2)\) 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 \( 1 + (1.32 - 2.29i)T \)
good2 \( 1 + (3.03 + 1.75i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (1.07 - 0.620i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (6.07 + 3.50i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-5.82 - 10.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 4.52iT - 289T^{2} \)
19 \( 1 - 16.2T + 361T^{2} \)
23 \( 1 + (-22.1 + 12.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-8.22 - 4.74i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (14.3 + 24.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 33.0T + 1.36e3T^{2} \)
41 \( 1 + (58.1 - 33.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-12.0 + 20.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (28.5 + 16.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 15.1iT - 2.80e3T^{2} \)
59 \( 1 + (80.0 - 46.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-28.7 + 49.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (7.58 + 13.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 70.5iT - 5.04e3T^{2} \)
73 \( 1 + 76.7T + 5.32e3T^{2} \)
79 \( 1 + (63.6 - 110. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-64.3 - 37.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 + (-11.5 + 20.0i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.78023130819942324311901224037, −9.855118612460123408622961557761, −9.142930117181752947898686255176, −8.453032944336904122891051177515, −7.54468262833688306399549026826, −6.70208919253633489908848156565, −5.25302849332820889673189761224, −3.59764433187187783202695359165, −2.66416269154861067556880969531, −1.36968546415118374133516800578, 0.14595998741563585656282962726, 1.48334574710289713997365846751, 3.28158235845773236142601134972, 5.01006925421146294634059233962, 5.98029717854416214374212808078, 7.01239447621786025600003894986, 7.67546134831878851024999568033, 8.434365833914671719771002535817, 9.232674923397905177045741732262, 10.20815480436580281451614605197

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