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Properties

Label 2-561-11.9-c1-0-29
Degree $2$
Conductor $561$
Sign $-0.307 - 0.951i$
Analytic cond. $4.47960$
Root an. cond. $2.11650$
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  + (0.768 − 2.36i)2-s + (−0.809 + 0.587i)3-s + (−3.38 − 2.45i)4-s + (−1.12 − 3.44i)5-s + (0.768 + 2.36i)6-s + (0.586 + 0.426i)7-s + (−4.38 + 3.18i)8-s + (0.309 − 0.951i)9-s − 9.01·10-s + (−1.82 − 2.77i)11-s + 4.18·12-s + (−1.35 + 4.16i)13-s + (1.45 − 1.05i)14-s + (2.93 + 2.13i)15-s + (1.58 + 4.87i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (0.543 − 1.67i)2-s + (−0.467 + 0.339i)3-s + (−1.69 − 1.22i)4-s + (−0.500 − 1.54i)5-s + (0.313 + 0.965i)6-s + (0.221 + 0.161i)7-s + (−1.55 + 1.12i)8-s + (0.103 − 0.317i)9-s − 2.85·10-s + (−0.548 − 0.835i)11-s + 1.20·12-s + (−0.375 + 1.15i)13-s + (0.389 − 0.283i)14-s + (0.757 + 0.550i)15-s + (0.396 + 1.21i)16-s + (0.0749 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(561\)    =    \(3 \cdot 11 \cdot 17\)
Sign: $-0.307 - 0.951i$
Analytic conductor: \(4.47960\)
Root analytic conductor: \(2.11650\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{561} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 561,\ (\ :1/2),\ -0.307 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.523291 + 0.719010i\)
\(L(\frac12)\) \(\approx\) \(0.523291 + 0.719010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (1.82 + 2.77i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.768 + 2.36i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (1.12 + 3.44i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.586 - 0.426i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.35 - 4.16i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (-2.28 + 1.65i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.17T + 23T^{2} \)
29 \( 1 + (6.25 + 4.54i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.24 - 3.82i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.56 - 4.04i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.03 + 4.38i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.72T + 43T^{2} \)
47 \( 1 + (-6.13 + 4.45i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.25 - 10.0i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.83 + 5.69i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.23 + 6.89i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 2.28T + 67T^{2} \)
71 \( 1 + (3.05 + 9.39i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.68 + 3.40i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-5.13 + 15.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.66 + 11.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-4.13 + 12.7i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.43644728716288492071920659670, −9.277205041867625981973105246106, −9.001106664431261383110640745184, −7.73180747894541649462502488849, −5.85582620994649587845152554564, −4.92979504809587451118194183951, −4.43572756008635467751162123993, −3.37135476347532012560305997282, −1.78391850425068584558173269921, −0.45614265746161942228537972035, 2.83991873393696022780593560774, 4.08947598217412316844715361550, 5.24876767112695645853628902016, 5.99768710570173246657617414770, 7.02160613042597161247329621439, 7.55468767433910341431944136360, 7.915516923920451083173730382895, 9.545499487471059410336632808122, 10.58586883717248796874068245290, 11.33568921084560326574534301032

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