Properties

Label 2-550-1.1-c1-0-14
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s − 5·7-s + 8-s − 2·9-s + 11-s − 12-s − 2·13-s − 5·14-s + 16-s − 3·17-s − 2·18-s − 7·19-s + 5·21-s + 22-s + 6·23-s − 24-s − 2·26-s + 5·27-s − 5·28-s − 3·29-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 − 1.88·7-s + 0.353·8-s − 2/3·9-s + 0.301·11-s − 0.288·12-s − 0.554·13-s − 1.33·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 1.60·19-s + 1.09·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s − 0.392·26-s + 0.962·27-s − 0.944·28-s − 0.557·29-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: \(1\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−10.59296143170299319105275536153, −9.515625676781913720459205161908, −8.766840852787028026444155939270, −7.20457953717496815600620253355, −6.43810722624715010093850437821, −5.90185767008403089888905192216, −4.68830133406592764908928362464, −3.51902803001209100282256024354, −2.52121766859740692142668949704, 0, 2.52121766859740692142668949704, 3.51902803001209100282256024354, 4.68830133406592764908928362464, 5.90185767008403089888905192216, 6.43810722624715010093850437821, 7.20457953717496815600620253355, 8.766840852787028026444155939270, 9.515625676781913720459205161908, 10.59296143170299319105275536153

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