Properties

Label 2-5577-1.1-c1-0-119
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.45·2-s + 3-s + 4.00·4-s + 4.42·5-s − 2.45·6-s + 1.04·7-s − 4.91·8-s + 9-s − 10.8·10-s + 11-s + 4.00·12-s − 2.55·14-s + 4.42·15-s + 4.03·16-s + 1.91·17-s − 2.45·18-s − 2.86·19-s + 17.7·20-s + 1.04·21-s − 2.45·22-s − 2.43·23-s − 4.91·24-s + 14.5·25-s + 27-s + 4.17·28-s − 7.02·29-s − 10.8·30-s + ⋯
L(s)  = 1  − 1.73·2-s + 0.577·3-s + 2.00·4-s + 1.97·5-s − 1.00·6-s + 0.393·7-s − 1.73·8-s + 0.333·9-s − 3.42·10-s + 0.301·11-s + 1.15·12-s − 0.682·14-s + 1.14·15-s + 1.00·16-s + 0.464·17-s − 0.577·18-s − 0.658·19-s + 3.95·20-s + 0.227·21-s − 0.522·22-s − 0.506·23-s − 1.00·24-s + 2.90·25-s + 0.192·27-s + 0.789·28-s − 1.30·29-s − 1.97·30-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

\(L(1)\) \(\approx\) \(1.844441295\)
\(L(\frac12)\) \(\approx\) \(1.844441295\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.45T + 2T^{2} \)
5 \( 1 - 4.42T + 5T^{2} \)
7 \( 1 - 1.04T + 7T^{2} \)
17 \( 1 - 1.91T + 17T^{2} \)
19 \( 1 + 2.86T + 19T^{2} \)
23 \( 1 + 2.43T + 23T^{2} \)
29 \( 1 + 7.02T + 29T^{2} \)
31 \( 1 - 3.56T + 31T^{2} \)
37 \( 1 - 8.30T + 37T^{2} \)
41 \( 1 + 1.82T + 41T^{2} \)
43 \( 1 + 6.12T + 43T^{2} \)
47 \( 1 - 8.92T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 2.69T + 61T^{2} \)
67 \( 1 + 7.53T + 67T^{2} \)
71 \( 1 - 2.65T + 71T^{2} \)
73 \( 1 + 3.87T + 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 - 5.54T + 83T^{2} \)
89 \( 1 - 9.19T + 89T^{2} \)
97 \( 1 - 0.785T + 97T^{2} \)
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

−8.279963037920507689489932887056, −7.74005327618980820312085314659, −6.80643913819230916493052536292, −6.29582741763634376293933312358, −5.59198458684649714091896060906, −4.57330514557006212673098198650, −3.17981414679422297333456161768, −2.18617795276830509954816799400, −1.86577811827156771308689382576, −0.959477417927984261982753351920, 0.959477417927984261982753351920, 1.86577811827156771308689382576, 2.18617795276830509954816799400, 3.17981414679422297333456161768, 4.57330514557006212673098198650, 5.59198458684649714091896060906, 6.29582741763634376293933312358, 6.80643913819230916493052536292, 7.74005327618980820312085314659, 8.279963037920507689489932887056

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