Properties

Label 2-4840-1.1-c1-0-17
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  − 3.03·3-s − 5-s − 0.865·7-s + 6.20·9-s + 4.43·13-s + 3.03·15-s + 0.550·17-s + 0.258·19-s + 2.62·21-s + 7.92·23-s + 25-s − 9.72·27-s + 9.36·29-s − 5.31·31-s + 0.865·35-s + 6.38·37-s − 13.4·39-s − 8.74·41-s − 6.38·43-s − 6.20·45-s + 0.823·47-s − 6.25·49-s − 1.66·51-s + 2.40·53-s − 0.785·57-s − 13.5·59-s + 6.60·61-s + ⋯
L(s)  = 1  − 1.75·3-s − 0.447·5-s − 0.327·7-s + 2.06·9-s + 1.23·13-s + 0.783·15-s + 0.133·17-s + 0.0594·19-s + 0.573·21-s + 1.65·23-s + 0.200·25-s − 1.87·27-s + 1.73·29-s − 0.954·31-s + 0.146·35-s + 1.05·37-s − 2.15·39-s − 1.36·41-s − 0.973·43-s − 0.925·45-s + 0.120·47-s − 0.892·49-s − 0.233·51-s + 0.329·53-s − 0.104·57-s − 1.75·59-s + 0.846·61-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

\(L(1)\) \(\approx\) \(0.9069773373\)
\(L(\frac12)\) \(\approx\) \(0.9069773373\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.03T + 3T^{2} \)
7 \( 1 + 0.865T + 7T^{2} \)
13 \( 1 - 4.43T + 13T^{2} \)
17 \( 1 - 0.550T + 17T^{2} \)
19 \( 1 - 0.258T + 19T^{2} \)
23 \( 1 - 7.92T + 23T^{2} \)
29 \( 1 - 9.36T + 29T^{2} \)
31 \( 1 + 5.31T + 31T^{2} \)
37 \( 1 - 6.38T + 37T^{2} \)
41 \( 1 + 8.74T + 41T^{2} \)
43 \( 1 + 6.38T + 43T^{2} \)
47 \( 1 - 0.823T + 47T^{2} \)
53 \( 1 - 2.40T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 - 6.60T + 61T^{2} \)
67 \( 1 - 7.17T + 67T^{2} \)
71 \( 1 - 2.79T + 71T^{2} \)
73 \( 1 - 2.89T + 73T^{2} \)
79 \( 1 - 5.91T + 79T^{2} \)
83 \( 1 - 7.09T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + 7.83T + 97T^{2} \)
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

−8.228299752436168054612116328789, −7.29978877927617020848546496252, −6.53398573691977923489887717919, −6.30180535820167430583210329466, −5.25195205143203521433496674521, −4.85200356230867401601817037112, −3.89572781210496752852367310317, −3.07070065802073219063808463281, −1.43779069459350104660452834953, −0.63248701056540259643557922289, 0.63248701056540259643557922289, 1.43779069459350104660452834953, 3.07070065802073219063808463281, 3.89572781210496752852367310317, 4.85200356230867401601817037112, 5.25195205143203521433496674521, 6.30180535820167430583210329466, 6.53398573691977923489887717919, 7.29978877927617020848546496252, 8.228299752436168054612116328789

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