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Properties

Label 2-567-9.5-c2-0-15
Degree $2$
Conductor $567$
Sign $0.984 - 0.173i$
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  + (−1.97 + 1.13i)2-s + (0.593 − 1.02i)4-s + (−4.28 − 2.47i)5-s + (−1.32 − 2.29i)7-s − 6.40i·8-s + 11.2·10-s + (−17.4 + 10.0i)11-s + (−12.9 + 22.3i)13-s + (5.21 + 3.01i)14-s + (9.66 + 16.7i)16-s − 6.60i·17-s + 5.53·19-s + (−5.08 + 2.93i)20-s + (22.9 − 39.8i)22-s + (−12.5 − 7.26i)23-s + ⋯
L(s)  = 1  + (−0.986 + 0.569i)2-s + (0.148 − 0.256i)4-s + (−0.856 − 0.494i)5-s + (−0.188 − 0.327i)7-s − 0.800i·8-s + 1.12·10-s + (−1.58 + 0.917i)11-s + (−0.994 + 1.72i)13-s + (0.372 + 0.215i)14-s + (0.604 + 1.04i)16-s − 0.388i·17-s + 0.291·19-s + (−0.254 + 0.146i)20-s + (1.04 − 1.80i)22-s + (−0.547 − 0.315i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(15.4496\)
Root analytic conductor: \(3.93060\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4168183331\)
\(L(\frac12)\) \(\approx\) \(0.4168183331\)
\(L(2)\) 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 \( 1 + (1.97 - 1.13i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (4.28 + 2.47i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (17.4 - 10.0i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (12.9 - 22.3i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 6.60iT - 289T^{2} \)
19 \( 1 - 5.53T + 361T^{2} \)
23 \( 1 + (12.5 + 7.26i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-15.1 + 8.74i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-22.6 + 39.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 40.8T + 1.36e3T^{2} \)
41 \( 1 + (26.3 + 15.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-26.7 - 46.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-24.1 + 13.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 6.45iT - 2.80e3T^{2} \)
59 \( 1 + (-37.0 - 21.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (17.7 + 30.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-41.8 + 72.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 113. iT - 5.04e3T^{2} \)
73 \( 1 + 25.5T + 5.32e3T^{2} \)
79 \( 1 + (40.9 + 70.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-24.9 + 14.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 5.92iT - 7.92e3T^{2} \)
97 \( 1 + (-47.2 - 81.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.04715847962103616007528452410, −9.731947109044639079724822306059, −8.681386383492307761148887167631, −7.70177185565394203595368728229, −7.46468209823706955667307603541, −6.38973403513994660780516954999, −4.74816980070062317974020362491, −4.16169079870618370087214371598, −2.38820043492932137837293277482, −0.41832076167498818772226271176, 0.60393194928536964399107087163, 2.59806199578636377570729556635, 3.21706714443798329407269871154, 5.06148151076729228053199128074, 5.78343141282735512727213640458, 7.45411507318786832326292926532, 8.003429346664350148238575557657, 8.635838326426156333751312191171, 9.971816482824299305065526776897, 10.41112161517349395881135326859

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