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Properties

Label 2-567-21.5-c1-0-21
Degree $2$
Conductor $567$
Sign $0.997 - 0.0633i$
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  + (1.5 + 0.866i)2-s + (0.5 + 0.866i)4-s + (1.5 − 2.59i)5-s + (0.5 + 2.59i)7-s − 1.73i·8-s + (4.5 − 2.59i)10-s + (1.5 − 0.866i)11-s − 1.73i·13-s + (−1.5 + 4.33i)14-s + (2.49 − 4.33i)16-s + (1.5 + 2.59i)17-s + (−4.5 − 2.59i)19-s + 3.00·20-s + 3·22-s + (4.5 + 2.59i)23-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s + (0.250 + 0.433i)4-s + (0.670 − 1.16i)5-s + (0.188 + 0.981i)7-s − 0.612i·8-s + (1.42 − 0.821i)10-s + (0.452 − 0.261i)11-s − 0.480i·13-s + (−0.400 + 1.15i)14-s + (0.624 − 1.08i)16-s + (0.363 + 0.630i)17-s + (−1.03 − 0.596i)19-s + 0.670·20-s + 0.639·22-s + (0.938 + 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.997 - 0.0633i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.997 - 0.0633i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.78153 + 0.0881697i\)
\(L(\frac12)\) \(\approx\) \(2.78153 + 0.0881697i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good2 \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.5 + 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.5 - 2.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.19iT - 29T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.5 - 4.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12 + 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (4.5 - 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15T + 83T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.73iT - 97T^{2} \)
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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

−10.83612539544925063462619053251, −9.641543751662888161345136036094, −8.950563405968829149034967475248, −8.177470631215673143637332892235, −6.73591855180941546792822897588, −5.90683315443345150031852315037, −5.22630748811403514302725045696, −4.54559088279150914636302971862, −3.14008897957664352112289749253, −1.42296098956808189743847050329, 1.88282122431754066292592030799, 2.98111872008725473374828761447, 3.98099744189407645154422530661, 4.85339783024823087286445123010, 6.15738775980917794869699719771, 6.85760792396199183012022457743, 7.904104209556466368867919522013, 9.203138542373742024064459302700, 10.34941416175605210783341552978, 10.73735800302809156860829307937

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