Properties

Label 2-5577-1.1-c1-0-219
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.584·2-s + 3-s − 1.65·4-s + 1.95·5-s + 0.584·6-s − 1.51·7-s − 2.13·8-s + 9-s + 1.14·10-s − 11-s − 1.65·12-s − 0.883·14-s + 1.95·15-s + 2.06·16-s + 3.44·17-s + 0.584·18-s − 5.09·19-s − 3.23·20-s − 1.51·21-s − 0.584·22-s − 0.701·23-s − 2.13·24-s − 1.18·25-s + 27-s + 2.50·28-s + 5.04·29-s + 1.14·30-s + ⋯
L(s)  = 1  + 0.413·2-s + 0.577·3-s − 0.829·4-s + 0.873·5-s + 0.238·6-s − 0.571·7-s − 0.756·8-s + 0.333·9-s + 0.361·10-s − 0.301·11-s − 0.478·12-s − 0.236·14-s + 0.504·15-s + 0.516·16-s + 0.834·17-s + 0.137·18-s − 1.16·19-s − 0.724·20-s − 0.329·21-s − 0.124·22-s − 0.146·23-s − 0.436·24-s − 0.236·25-s + 0.192·27-s + 0.473·28-s + 0.937·29-s + 0.208·30-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 \( 1 - 0.584T + 2T^{2} \)
5 \( 1 - 1.95T + 5T^{2} \)
7 \( 1 + 1.51T + 7T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 + 5.09T + 19T^{2} \)
23 \( 1 + 0.701T + 23T^{2} \)
29 \( 1 - 5.04T + 29T^{2} \)
31 \( 1 + 7.26T + 31T^{2} \)
37 \( 1 - 2.08T + 37T^{2} \)
41 \( 1 + 1.03T + 41T^{2} \)
43 \( 1 + 5.48T + 43T^{2} \)
47 \( 1 + 3.75T + 47T^{2} \)
53 \( 1 + 0.502T + 53T^{2} \)
59 \( 1 - 2.65T + 59T^{2} \)
61 \( 1 + 5.38T + 61T^{2} \)
67 \( 1 - 8.47T + 67T^{2} \)
71 \( 1 - 2.31T + 71T^{2} \)
73 \( 1 + 2.21T + 73T^{2} \)
79 \( 1 + 5.54T + 79T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 - 1.80T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.024141478794055516398947909333, −6.96484247772884881127287645414, −6.21192791195489747490143496887, −5.60633526958830095981297042052, −4.87498941031878733439675333698, −4.02290741327592396094541255163, −3.31318899305983818402999414100, −2.52766947556852410395817094310, −1.49070717794361119798046986645, 0, 1.49070717794361119798046986645, 2.52766947556852410395817094310, 3.31318899305983818402999414100, 4.02290741327592396094541255163, 4.87498941031878733439675333698, 5.60633526958830095981297042052, 6.21192791195489747490143496887, 6.96484247772884881127287645414, 8.024141478794055516398947909333

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