Properties

Label 2-4840-1.1-c1-0-89
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.614·3-s + 5-s − 2.24·7-s − 2.62·9-s + 5.19·13-s + 0.614·15-s − 3.22·17-s − 7.19·19-s − 1.38·21-s + 5.11·23-s + 25-s − 3.45·27-s + 9.61·29-s + 0.491·31-s − 2.24·35-s + 0.350·37-s + 3.19·39-s − 4.42·41-s − 10.9·43-s − 2.62·45-s − 1.03·47-s − 1.95·49-s − 1.98·51-s + 3.75·53-s − 4.42·57-s + 12.7·59-s − 6.38·61-s + ⋯
L(s)  = 1  + 0.355·3-s + 0.447·5-s − 0.848·7-s − 0.873·9-s + 1.44·13-s + 0.158·15-s − 0.783·17-s − 1.65·19-s − 0.301·21-s + 1.06·23-s + 0.200·25-s − 0.665·27-s + 1.78·29-s + 0.0882·31-s − 0.379·35-s + 0.0576·37-s + 0.511·39-s − 0.690·41-s − 1.67·43-s − 0.390·45-s − 0.150·47-s − 0.279·49-s − 0.278·51-s + 0.515·53-s − 0.586·57-s + 1.65·59-s − 0.818·61-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: \(1\)
Selberg data: \((2,\ 4840,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good3 \( 1 - 0.614T + 3T^{2} \)
7 \( 1 + 2.24T + 7T^{2} \)
13 \( 1 - 5.19T + 13T^{2} \)
17 \( 1 + 3.22T + 17T^{2} \)
19 \( 1 + 7.19T + 19T^{2} \)
23 \( 1 - 5.11T + 23T^{2} \)
29 \( 1 - 9.61T + 29T^{2} \)
31 \( 1 - 0.491T + 31T^{2} \)
37 \( 1 - 0.350T + 37T^{2} \)
41 \( 1 + 4.42T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 1.03T + 47T^{2} \)
53 \( 1 - 3.75T + 53T^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 6.38T + 61T^{2} \)
67 \( 1 + 8.44T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 - 2.13T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 - 9.90T + 89T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.290423004444549360962734998004, −6.93411371138750262170456244985, −6.42987597082227600940222572622, −5.97200125207206070418377572673, −4.95680075765627903543513527458, −4.06553203345657894910602162357, −3.16405449445578732334569760020, −2.59117423742805720460575045348, −1.44676036649645738241952603334, 0, 1.44676036649645738241952603334, 2.59117423742805720460575045348, 3.16405449445578732334569760020, 4.06553203345657894910602162357, 4.95680075765627903543513527458, 5.97200125207206070418377572673, 6.42987597082227600940222572622, 6.93411371138750262170456244985, 8.290423004444549360962734998004

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