Properties

Label 2-550-1.1-c1-0-4
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 + 4·13-s + 3·14-s + 16-s + 3·17-s − 2·18-s − 5·19-s − 3·21-s + 22-s + 4·23-s − 24-s + 4·26-s + 5·27-s + 3·28-s + 5·29-s + 7·31-s + 32-s − 33-s + 3·34-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 + 1.10·13-s + 0.801·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 1.14·19-s − 0.654·21-s + 0.213·22-s + 0.834·23-s − 0.204·24-s + 0.784·26-s + 0.962·27-s + 0.566·28-s + 0.928·29-s + 1.25·31-s + 0.176·32-s − 0.174·33-s + 0.514·34-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

\(L(1)\) \(\approx\) \(2.074912862\)
\(L(\frac12)\) \(\approx\) \(2.074912862\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−10.87859972852879920732148596028, −10.41613231225318065197074702698, −8.717961421562585762296647152002, −8.242907688677664015983227610335, −6.89040237899544266304191025391, −6.04641270241765746429835400460, −5.18711436468138586194772102374, −4.31835852121277184532189905403, −3.01207625761226941022973660182, −1.40662093843722554519102852406, 1.40662093843722554519102852406, 3.01207625761226941022973660182, 4.31835852121277184532189905403, 5.18711436468138586194772102374, 6.04641270241765746429835400460, 6.89040237899544266304191025391, 8.242907688677664015983227610335, 8.717961421562585762296647152002, 10.41613231225318065197074702698, 10.87859972852879920732148596028

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