Properties

Label 2-5577-1.1-c1-0-109
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  − 0.849·2-s − 3-s − 1.27·4-s − 2.48·5-s + 0.849·6-s + 1.35·7-s + 2.78·8-s + 9-s + 2.11·10-s − 11-s + 1.27·12-s − 1.14·14-s + 2.48·15-s + 0.194·16-s − 2.42·17-s − 0.849·18-s − 0.806·19-s + 3.17·20-s − 1.35·21-s + 0.849·22-s − 2.92·23-s − 2.78·24-s + 1.17·25-s − 27-s − 1.72·28-s + 8.28·29-s − 2.11·30-s + ⋯
L(s)  = 1  − 0.600·2-s − 0.577·3-s − 0.639·4-s − 1.11·5-s + 0.346·6-s + 0.510·7-s + 0.984·8-s + 0.333·9-s + 0.667·10-s − 0.301·11-s + 0.369·12-s − 0.306·14-s + 0.641·15-s + 0.0485·16-s − 0.587·17-s − 0.200·18-s − 0.185·19-s + 0.710·20-s − 0.294·21-s + 0.181·22-s − 0.609·23-s − 0.568·24-s + 0.235·25-s − 0.192·27-s − 0.326·28-s + 1.53·29-s − 0.385·30-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 \( 1 + 0.849T + 2T^{2} \)
5 \( 1 + 2.48T + 5T^{2} \)
7 \( 1 - 1.35T + 7T^{2} \)
17 \( 1 + 2.42T + 17T^{2} \)
19 \( 1 + 0.806T + 19T^{2} \)
23 \( 1 + 2.92T + 23T^{2} \)
29 \( 1 - 8.28T + 29T^{2} \)
31 \( 1 - 2.29T + 31T^{2} \)
37 \( 1 + 5.14T + 37T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 + 7.56T + 43T^{2} \)
47 \( 1 - 7.73T + 47T^{2} \)
53 \( 1 - 7.15T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 1.67T + 61T^{2} \)
67 \( 1 + 7.05T + 67T^{2} \)
71 \( 1 - 9.88T + 71T^{2} \)
73 \( 1 - 3.70T + 73T^{2} \)
79 \( 1 + 1.65T + 79T^{2} \)
83 \( 1 + 2.85T + 83T^{2} \)
89 \( 1 - 1.84T + 89T^{2} \)
97 \( 1 + 0.284T + 97T^{2} \)
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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

−7.996653011366227426258811703534, −7.22466718878483064931880175595, −6.55447280661159950872633549984, −5.46681875932353935734201082148, −4.76327952570232624441621744776, −4.25603747428403513371741761515, −3.47980130970615359816006590004, −2.11406696296252633151124575644, −0.924512800842892023118110063063, 0, 0.924512800842892023118110063063, 2.11406696296252633151124575644, 3.47980130970615359816006590004, 4.25603747428403513371741761515, 4.76327952570232624441621744776, 5.46681875932353935734201082148, 6.55447280661159950872633549984, 7.22466718878483064931880175595, 7.996653011366227426258811703534

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