Properties

Label 2-4840-1.1-c1-0-45
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  + 2.33·3-s − 5-s + 0.399·7-s + 2.43·9-s + 2.43·13-s − 2.33·15-s + 3.63·19-s + 0.932·21-s − 1.02·23-s + 25-s − 1.30·27-s + 4·29-s + 3.23·31-s − 0.399·35-s − 4.87·37-s + 5.68·39-s + 0.0922·41-s + 7.14·43-s − 2.43·45-s + 9.21·47-s − 6.84·49-s + 9.10·53-s + 8.48·57-s − 7.68·59-s + 6.57·61-s + 0.975·63-s − 2.43·65-s + ⋯
L(s)  = 1  + 1.34·3-s − 0.447·5-s + 0.151·7-s + 0.813·9-s + 0.676·13-s − 0.602·15-s + 0.835·19-s + 0.203·21-s − 0.213·23-s + 0.200·25-s − 0.251·27-s + 0.742·29-s + 0.581·31-s − 0.0675·35-s − 0.802·37-s + 0.911·39-s + 0.0144·41-s + 1.08·43-s − 0.363·45-s + 1.34·47-s − 0.977·49-s + 1.25·53-s + 1.12·57-s − 1.00·59-s + 0.841·61-s + 0.122·63-s − 0.302·65-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

\(L(1)\) \(\approx\) \(3.208793818\)
\(L(\frac12)\) \(\approx\) \(3.208793818\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.33T + 3T^{2} \)
7 \( 1 - 0.399T + 7T^{2} \)
13 \( 1 - 2.43T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 3.63T + 19T^{2} \)
23 \( 1 + 1.02T + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 3.23T + 31T^{2} \)
37 \( 1 + 4.87T + 37T^{2} \)
41 \( 1 - 0.0922T + 41T^{2} \)
43 \( 1 - 7.14T + 43T^{2} \)
47 \( 1 - 9.21T + 47T^{2} \)
53 \( 1 - 9.10T + 53T^{2} \)
59 \( 1 + 7.68T + 59T^{2} \)
61 \( 1 - 6.57T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 - 5.28T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + 1.67T + 79T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 - 11.5T + 97T^{2} \)
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

−8.205543550792376556431011759295, −7.78820622234746806504900921547, −7.07037148544075301079796740035, −6.18830471941882506286560591678, −5.26377254156729085834048251869, −4.30814870887157937736509857304, −3.61550295249910072824576067230, −2.95872625815050918214294982730, −2.10127485311860860386254508760, −0.955099733925397507986936271474, 0.955099733925397507986936271474, 2.10127485311860860386254508760, 2.95872625815050918214294982730, 3.61550295249910072824576067230, 4.30814870887157937736509857304, 5.26377254156729085834048251869, 6.18830471941882506286560591678, 7.07037148544075301079796740035, 7.78820622234746806504900921547, 8.205543550792376556431011759295

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