Properties

Label 2-5577-1.1-c1-0-132
Degree $2$
Conductor $5577$
Sign $-1$
Analytic cond. $44.5325$
Root an. cond. $6.67327$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.33·2-s − 3-s − 0.226·4-s − 3.31·5-s − 1.33·6-s − 1.53·7-s − 2.96·8-s + 9-s − 4.41·10-s − 11-s + 0.226·12-s − 2.04·14-s + 3.31·15-s − 3.49·16-s + 5.96·17-s + 1.33·18-s + 6.46·19-s + 0.749·20-s + 1.53·21-s − 1.33·22-s + 5.32·23-s + 2.96·24-s + 5.97·25-s − 27-s + 0.347·28-s − 0.191·29-s + 4.41·30-s + ⋯
L(s)  = 1  + 0.941·2-s − 0.577·3-s − 0.113·4-s − 1.48·5-s − 0.543·6-s − 0.579·7-s − 1.04·8-s + 0.333·9-s − 1.39·10-s − 0.301·11-s + 0.0653·12-s − 0.546·14-s + 0.855·15-s − 0.874·16-s + 1.44·17-s + 0.313·18-s + 1.48·19-s + 0.167·20-s + 0.334·21-s − 0.283·22-s + 1.10·23-s + 0.605·24-s + 1.19·25-s − 0.192·27-s + 0.0656·28-s − 0.0355·29-s + 0.805·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5577\)    =    \(3 \cdot 11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(44.5325\)
Root analytic conductor: \(6.67327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5577,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - 1.33T + 2T^{2} \)
5 \( 1 + 3.31T + 5T^{2} \)
7 \( 1 + 1.53T + 7T^{2} \)
17 \( 1 - 5.96T + 17T^{2} \)
19 \( 1 - 6.46T + 19T^{2} \)
23 \( 1 - 5.32T + 23T^{2} \)
29 \( 1 + 0.191T + 29T^{2} \)
31 \( 1 + 9.38T + 31T^{2} \)
37 \( 1 - 9.26T + 37T^{2} \)
41 \( 1 - 0.250T + 41T^{2} \)
43 \( 1 + 5.16T + 43T^{2} \)
47 \( 1 + 4.36T + 47T^{2} \)
53 \( 1 - 7.13T + 53T^{2} \)
59 \( 1 + 9.80T + 59T^{2} \)
61 \( 1 - 8.46T + 61T^{2} \)
67 \( 1 + 6.81T + 67T^{2} \)
71 \( 1 + 7.41T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 6.63T + 79T^{2} \)
83 \( 1 + 9.84T + 83T^{2} \)
89 \( 1 - 5.37T + 89T^{2} \)
97 \( 1 + 7.24T + 97T^{2} \)
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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

−7.57137166454170156318025239962, −7.13539048197596280080617894543, −6.15564889249007486997685310430, −5.39157893735887924071960477477, −4.95800901347098149843293190699, −4.04835640526570916764434614723, −3.40035198354971759339834055681, −2.98200943124251675413052613351, −1.03864275584036559441033612239, 0, 1.03864275584036559441033612239, 2.98200943124251675413052613351, 3.40035198354971759339834055681, 4.04835640526570916764434614723, 4.95800901347098149843293190699, 5.39157893735887924071960477477, 6.15564889249007486997685310430, 7.13539048197596280080617894543, 7.57137166454170156318025239962

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