Properties

Label 2-550-1.1-c1-0-2
Degree $2$
Conductor $550$
Sign $1$
Analytic cond. $4.39177$
Root an. cond. $2.09565$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s − 2·9-s + 11-s + 12-s + 6·13-s + 3·14-s + 16-s + 7·17-s + 2·18-s + 5·19-s − 3·21-s − 22-s + 6·23-s − 24-s − 6·26-s − 5·27-s − 3·28-s + 5·29-s − 3·31-s − 32-s + 33-s − 7·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.301·11-s + 0.288·12-s + 1.66·13-s + 0.801·14-s + 1/4·16-s + 1.69·17-s + 0.471·18-s + 1.14·19-s − 0.654·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s − 1.17·26-s − 0.962·27-s − 0.566·28-s + 0.928·29-s − 0.538·31-s − 0.176·32-s + 0.174·33-s − 1.20·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.216011166\)
\(L(\frac12)\) \(\approx\) \(1.216011166\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
11 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−10.62761614296171945475675158355, −9.696869758195084295021589593937, −9.051313158030191365780373658814, −8.319735201663290337491520314526, −7.35838560113530362823348866724, −6.32452856749704006618306134382, −5.51294812234663797155432190910, −3.46996281594193181675719351221, −3.07694177217990984214438626883, −1.14442070453994046878747912184, 1.14442070453994046878747912184, 3.07694177217990984214438626883, 3.46996281594193181675719351221, 5.51294812234663797155432190910, 6.32452856749704006618306134382, 7.35838560113530362823348866724, 8.319735201663290337491520314526, 9.051313158030191365780373658814, 9.696869758195084295021589593937, 10.62761614296171945475675158355

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