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Properties

Label 2-561-11.4-c1-0-1
Degree $2$
Conductor $561$
Sign $0.456 - 0.889i$
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  + (−0.695 − 0.505i)2-s + (0.309 − 0.951i)3-s + (−0.389 − 1.19i)4-s + (−1.95 + 1.42i)5-s + (−0.695 + 0.505i)6-s + (0.322 + 0.992i)7-s + (−0.866 + 2.66i)8-s + (−0.809 − 0.587i)9-s + 2.07·10-s + (−2.28 − 2.40i)11-s − 1.26·12-s + (3.17 + 2.30i)13-s + (0.277 − 0.852i)14-s + (0.747 + 2.30i)15-s + (−0.0915 + 0.0664i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.491 − 0.357i)2-s + (0.178 − 0.549i)3-s + (−0.194 − 0.599i)4-s + (−0.875 + 0.635i)5-s + (−0.283 + 0.206i)6-s + (0.121 + 0.375i)7-s + (−0.306 + 0.942i)8-s + (−0.269 − 0.195i)9-s + 0.657·10-s + (−0.687 − 0.726i)11-s − 0.364·12-s + (0.881 + 0.640i)13-s + (0.0740 − 0.227i)14-s + (0.193 + 0.594i)15-s + (−0.0228 + 0.0166i)16-s + (−0.196 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(561\)    =    \(3 \cdot 11 \cdot 17\)
Sign: $0.456 - 0.889i$
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: \((2,\ 561,\ (\ :1/2),\ 0.456 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.422444 + 0.257908i\)
\(L(\frac12)\) \(\approx\) \(0.422444 + 0.257908i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.28 + 2.40i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (0.695 + 0.505i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (1.95 - 1.42i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.322 - 0.992i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.17 - 2.30i)T + (4.01 + 12.3i)T^{2} \)
19 \( 1 + (1.64 - 5.05i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.52T + 23T^{2} \)
29 \( 1 + (-2.55 - 7.87i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.92 - 5.75i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.07 + 6.38i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.16 + 3.59i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.286T + 43T^{2} \)
47 \( 1 + (1.22 - 3.75i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.61 - 4.80i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.477 - 1.47i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.29 - 4.57i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 + (11.3 - 8.24i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.25 + 10.0i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.830 - 0.603i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.93 - 2.86i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + (-2.66 - 1.93i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.69885614000784401616008806329, −10.39415385665705667367833335353, −8.871354098080842153841650848238, −8.473604377365200057749223613286, −7.53960208068761435696154806589, −6.34271061958056296245552301884, −5.60572775527624106657493111854, −4.07489500795773024801475969582, −2.84763294077703187611066716071, −1.51102139401650341857768648360, 0.33728235975775416217408226035, 2.82567295734890621430442237425, 4.20177532001775955700189961519, 4.52342451973636842596428471334, 6.14720672965759827966761975448, 7.40146863587589882064267186191, 8.179344030852128065964935624085, 8.494829953316559503075163322915, 9.687594076526421605710214356981, 10.34552532100912509127416116046

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