Properties

Label 2-5577-1.1-c1-0-137
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.37·2-s + 3-s − 0.111·4-s + 2.03·5-s − 1.37·6-s + 3.75·7-s + 2.90·8-s + 9-s − 2.79·10-s + 11-s − 0.111·12-s − 5.15·14-s + 2.03·15-s − 3.76·16-s + 3.59·17-s − 1.37·18-s + 5.32·19-s − 0.226·20-s + 3.75·21-s − 1.37·22-s + 3.72·23-s + 2.90·24-s − 0.866·25-s + 27-s − 0.417·28-s − 5.11·29-s − 2.79·30-s + ⋯
L(s)  = 1  − 0.971·2-s + 0.577·3-s − 0.0556·4-s + 0.909·5-s − 0.561·6-s + 1.41·7-s + 1.02·8-s + 0.333·9-s − 0.883·10-s + 0.301·11-s − 0.0321·12-s − 1.37·14-s + 0.524·15-s − 0.941·16-s + 0.871·17-s − 0.323·18-s + 1.22·19-s − 0.0506·20-s + 0.818·21-s − 0.292·22-s + 0.777·23-s + 0.592·24-s − 0.173·25-s + 0.192·27-s − 0.0789·28-s − 0.950·29-s − 0.510·30-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: \(0\)
Selberg data: \((2,\ 5577,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.283059195\)
\(L(\frac12)\) \(\approx\) \(2.283059195\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.37T + 2T^{2} \)
5 \( 1 - 2.03T + 5T^{2} \)
7 \( 1 - 3.75T + 7T^{2} \)
17 \( 1 - 3.59T + 17T^{2} \)
19 \( 1 - 5.32T + 19T^{2} \)
23 \( 1 - 3.72T + 23T^{2} \)
29 \( 1 + 5.11T + 29T^{2} \)
31 \( 1 + 9.37T + 31T^{2} \)
37 \( 1 - 11.2T + 37T^{2} \)
41 \( 1 - 2.06T + 41T^{2} \)
43 \( 1 - 9.52T + 43T^{2} \)
47 \( 1 + 2.02T + 47T^{2} \)
53 \( 1 + 0.421T + 53T^{2} \)
59 \( 1 - 1.76T + 59T^{2} \)
61 \( 1 - 9.16T + 61T^{2} \)
67 \( 1 + 1.58T + 67T^{2} \)
71 \( 1 + 9.70T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 + 4.46T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−8.147580796204911391059280907132, −7.55573986863869217782543136685, −7.27580644814871544665988125964, −5.85602562660117551228160169736, −5.32667592416332118808389803226, −4.52074483323360138862940705912, −3.66643856619294109497935251532, −2.45632492424363767045377520448, −1.60014536018718371074853290171, −1.04987029688691895975331642845, 1.04987029688691895975331642845, 1.60014536018718371074853290171, 2.45632492424363767045377520448, 3.66643856619294109497935251532, 4.52074483323360138862940705912, 5.32667592416332118808389803226, 5.85602562660117551228160169736, 7.27580644814871544665988125964, 7.55573986863869217782543136685, 8.147580796204911391059280907132

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