Properties

Label 2-4840-1.1-c1-0-24
Degree $2$
Conductor $4840$
Sign $1$
Analytic cond. $38.6475$
Root an. cond. $6.21671$
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  + 1.89·3-s − 5-s − 2.74·7-s + 0.574·9-s + 0.196·13-s − 1.89·15-s − 1.46·17-s − 2.68·19-s − 5.18·21-s + 1.81·23-s + 25-s − 4.58·27-s + 1.37·29-s + 5.90·31-s + 2.74·35-s + 4.57·37-s + 0.372·39-s + 1.23·41-s + 2.96·43-s − 0.574·45-s + 9.30·47-s + 0.511·49-s − 2.77·51-s + 2.69·53-s − 5.06·57-s + 14.2·59-s + 7.54·61-s + ⋯
L(s)  = 1  + 1.09·3-s − 0.447·5-s − 1.03·7-s + 0.191·9-s + 0.0545·13-s − 0.488·15-s − 0.356·17-s − 0.615·19-s − 1.13·21-s + 0.378·23-s + 0.200·25-s − 0.882·27-s + 0.256·29-s + 1.06·31-s + 0.463·35-s + 0.752·37-s + 0.0595·39-s + 0.192·41-s + 0.452·43-s − 0.0856·45-s + 1.35·47-s + 0.0731·49-s − 0.388·51-s + 0.370·53-s − 0.671·57-s + 1.84·59-s + 0.965·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4840\)    =    \(2^{3} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(38.6475\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.014069850\)
\(L(\frac12)\) \(\approx\) \(2.014069850\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good3 \( 1 - 1.89T + 3T^{2} \)
7 \( 1 + 2.74T + 7T^{2} \)
13 \( 1 - 0.196T + 13T^{2} \)
17 \( 1 + 1.46T + 17T^{2} \)
19 \( 1 + 2.68T + 19T^{2} \)
23 \( 1 - 1.81T + 23T^{2} \)
29 \( 1 - 1.37T + 29T^{2} \)
31 \( 1 - 5.90T + 31T^{2} \)
37 \( 1 - 4.57T + 37T^{2} \)
41 \( 1 - 1.23T + 41T^{2} \)
43 \( 1 - 2.96T + 43T^{2} \)
47 \( 1 - 9.30T + 47T^{2} \)
53 \( 1 - 2.69T + 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 - 7.54T + 61T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 + 6.87T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 - 1.87T + 79T^{2} \)
83 \( 1 + 5.79T + 83T^{2} \)
89 \( 1 - 17.3T + 89T^{2} \)
97 \( 1 + 18.3T + 97T^{2} \)
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

−8.382778618920176455267379640849, −7.67852785754176653397273149184, −6.87687664527645943444798064689, −6.29043521682432757196178999976, −5.35005738863988685650164487119, −4.22861370465898458402186587236, −3.70358216684842333118204036910, −2.81583212339917171171150035228, −2.30199485577314913186338598325, −0.71444935043033959757559397992, 0.71444935043033959757559397992, 2.30199485577314913186338598325, 2.81583212339917171171150035228, 3.70358216684842333118204036910, 4.22861370465898458402186587236, 5.35005738863988685650164487119, 6.29043521682432757196178999976, 6.87687664527645943444798064689, 7.67852785754176653397273149184, 8.382778618920176455267379640849

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