Properties

Label 2-5577-1.1-c1-0-88
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  − 0.409·2-s + 3-s − 1.83·4-s − 4.13·5-s − 0.409·6-s − 5.18·7-s + 1.56·8-s + 9-s + 1.69·10-s − 11-s − 1.83·12-s + 2.12·14-s − 4.13·15-s + 3.02·16-s − 0.488·17-s − 0.409·18-s + 0.446·19-s + 7.58·20-s − 5.18·21-s + 0.409·22-s + 5.50·23-s + 1.56·24-s + 12.1·25-s + 27-s + 9.49·28-s − 6.58·29-s + 1.69·30-s + ⋯
L(s)  = 1  − 0.289·2-s + 0.577·3-s − 0.916·4-s − 1.85·5-s − 0.167·6-s − 1.95·7-s + 0.554·8-s + 0.333·9-s + 0.535·10-s − 0.301·11-s − 0.529·12-s + 0.566·14-s − 1.06·15-s + 0.755·16-s − 0.118·17-s − 0.0964·18-s + 0.102·19-s + 1.69·20-s − 1.13·21-s + 0.0872·22-s + 1.14·23-s + 0.320·24-s + 2.42·25-s + 0.192·27-s + 1.79·28-s − 1.22·29-s + 0.309·30-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: $\chi_{5577} (1, \cdot )$
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 \( 1 + 0.409T + 2T^{2} \)
5 \( 1 + 4.13T + 5T^{2} \)
7 \( 1 + 5.18T + 7T^{2} \)
17 \( 1 + 0.488T + 17T^{2} \)
19 \( 1 - 0.446T + 19T^{2} \)
23 \( 1 - 5.50T + 23T^{2} \)
29 \( 1 + 6.58T + 29T^{2} \)
31 \( 1 + 1.52T + 31T^{2} \)
37 \( 1 - 7.87T + 37T^{2} \)
41 \( 1 - 8.78T + 41T^{2} \)
43 \( 1 + 4.57T + 43T^{2} \)
47 \( 1 + 5.42T + 47T^{2} \)
53 \( 1 - 8.66T + 53T^{2} \)
59 \( 1 + 5.60T + 59T^{2} \)
61 \( 1 + 0.855T + 61T^{2} \)
67 \( 1 - 3.87T + 67T^{2} \)
71 \( 1 - 8.49T + 71T^{2} \)
73 \( 1 + 5.95T + 73T^{2} \)
79 \( 1 + 8.91T + 79T^{2} \)
83 \( 1 + 0.457T + 83T^{2} \)
89 \( 1 - 4.68T + 89T^{2} \)
97 \( 1 - 2.06T + 97T^{2} \)
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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

−7.76598641079031621760655633547, −7.33911847254728015153048421381, −6.64197999534141382148289162105, −5.57807946789852243748171063541, −4.56267960886502387710842452379, −3.90221985717832957547559929245, −3.40505675446766046204309156637, −2.75358532228487045328198558265, −0.827590777756021901164862561951, 0, 0.827590777756021901164862561951, 2.75358532228487045328198558265, 3.40505675446766046204309156637, 3.90221985717832957547559929245, 4.56267960886502387710842452379, 5.57807946789852243748171063541, 6.64197999534141382148289162105, 7.33911847254728015153048421381, 7.76598641079031621760655633547

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