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Properties

Label 2-561-11.4-c1-0-22
Degree $2$
Conductor $561$
Sign $0.886 + 0.462i$
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  + (1.23 + 0.899i)2-s + (0.309 − 0.951i)3-s + (0.104 + 0.322i)4-s + (−0.0100 + 0.00726i)5-s + (1.23 − 0.899i)6-s + (−0.0399 − 0.122i)7-s + (0.784 − 2.41i)8-s + (−0.809 − 0.587i)9-s − 0.0189·10-s + (2.42 − 2.26i)11-s + 0.339·12-s + (−0.0708 − 0.0514i)13-s + (0.0610 − 0.188i)14-s + (0.00382 + 0.0117i)15-s + (3.69 − 2.68i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.874 + 0.635i)2-s + (0.178 − 0.549i)3-s + (0.0524 + 0.161i)4-s + (−0.00447 + 0.00325i)5-s + (0.505 − 0.367i)6-s + (−0.0150 − 0.0464i)7-s + (0.277 − 0.854i)8-s + (−0.269 − 0.195i)9-s − 0.00598·10-s + (0.730 − 0.683i)11-s + 0.0979·12-s + (−0.0196 − 0.0142i)13-s + (0.0163 − 0.0502i)14-s + (0.000986 + 0.00303i)15-s + (0.923 − 0.670i)16-s + (−0.196 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(561\)    =    \(3 \cdot 11 \cdot 17\)
Sign: $0.886 + 0.462i$
Analytic conductor: \(4.47960\)
Root analytic conductor: \(2.11650\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{561} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 561,\ (\ :1/2),\ 0.886 + 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32568 - 0.569702i\)
\(L(\frac12)\) \(\approx\) \(2.32568 - 0.569702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-2.42 + 2.26i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-1.23 - 0.899i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (0.0100 - 0.00726i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.0399 + 0.122i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.0708 + 0.0514i)T + (4.01 + 12.3i)T^{2} \)
19 \( 1 + (-1.02 + 3.14i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.09T + 23T^{2} \)
29 \( 1 + (0.456 + 1.40i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.65 - 1.92i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.67 + 5.16i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.05 - 9.39i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.20T + 43T^{2} \)
47 \( 1 + (3.85 - 11.8i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.06 + 2.95i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.00727 + 0.0223i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.558 - 0.405i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.91T + 67T^{2} \)
71 \( 1 + (4.87 - 3.53i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.97 - 12.2i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.68 - 4.13i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.45 - 3.23i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.111T + 89T^{2} \)
97 \( 1 + (-1.78 - 1.29i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.90398102275584461953314803754, −9.640844814072847680450126968073, −8.888504314985100338629955226069, −7.72216710553073276277661483680, −6.86451404189201204236614204187, −6.16391731726535070838844914724, −5.24084259404314919444628859831, −4.13695966244444010850317595717, −3.03775423171051441667551218598, −1.17435424452883612773474005412, 1.96378431738998098899133389090, 3.19028197958060607144648555186, 4.12570890704722359624403130467, 4.85692165172804967464388868215, 5.94876284526810274324174631696, 7.22556384466288054560362944870, 8.311814964489721869748943653834, 9.167005595591545954408010598227, 10.16544823002606781053157607173, 10.89719183206681797842062898104

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