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Properties

Label 2-550-11.3-c1-0-15
Degree $2$
Conductor $550$
Sign $0.0694 + 0.997i$
Analytic cond. $4.39177$
Root an. cond. $2.09565$
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  + (0.809 − 0.587i)2-s + (−0.118 − 0.363i)3-s + (0.309 − 0.951i)4-s + (−0.309 − 0.224i)6-s + (0.618 − 1.90i)7-s + (−0.309 − 0.951i)8-s + (2.30 − 1.67i)9-s + (0.309 + 3.30i)11-s − 0.381·12-s + (1 − 0.726i)13-s + (−0.618 − 1.90i)14-s + (−0.809 − 0.587i)16-s + (−0.5 − 0.363i)17-s + (0.881 − 2.71i)18-s + (−1.80 − 5.56i)19-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.0681 − 0.209i)3-s + (0.154 − 0.475i)4-s + (−0.126 − 0.0916i)6-s + (0.233 − 0.718i)7-s + (−0.109 − 0.336i)8-s + (0.769 − 0.559i)9-s + (0.0931 + 0.995i)11-s − 0.110·12-s + (0.277 − 0.201i)13-s + (−0.165 − 0.508i)14-s + (−0.202 − 0.146i)16-s + (−0.121 − 0.0881i)17-s + (0.207 − 0.639i)18-s + (−0.415 − 1.27i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $0.0694 + 0.997i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 0.0694 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48346 - 1.38374i\)
\(L(\frac12)\) \(\approx\) \(1.48346 - 1.38374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 \)
11 \( 1 + (-0.309 - 3.30i)T \)
good3 \( 1 + (0.118 + 0.363i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.618 + 1.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1 + 0.726i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.5 + 0.363i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.80 + 5.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.23T + 23T^{2} \)
29 \( 1 + (-1.38 + 4.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.61 - 1.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.14 + 3.52i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.73 - 5.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.56T + 43T^{2} \)
47 \( 1 + (-2 - 6.15i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.23 - 0.898i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.66 - 8.19i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2 - 1.45i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 + (-4.23 - 3.07i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.20 - 9.87i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.8 - 7.88i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.54 - 5.48i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.09T + 89T^{2} \)
97 \( 1 + (-5.78 + 4.20i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.68240029951816313189585725866, −9.864955679097213141233752707881, −9.049156231915101496080323045666, −7.59136092034050329540387865064, −6.96405351454224261806718022957, −5.97828629415476401509342232398, −4.55842477148126905892584541966, −4.08603171129867236669014394368, −2.53325479026692475600470341648, −1.10438334419333507652397650544, 1.92703481008193830128475094257, 3.43501888196232190351204212909, 4.43787659146710971130419304792, 5.52609474563702034630838509700, 6.20101147569022631086184850975, 7.38849177388159350924628745434, 8.295434183744741971952631117336, 9.034962132767148307916916069927, 10.28418101935943543927168791245, 11.02431154086569791553770394345

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