Properties

Label 2-4840-1.1-c1-0-60
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  − 2.96·3-s + 5-s − 1.02·7-s + 5.79·9-s + 0.504·13-s − 2.96·15-s + 3.93·17-s − 2.50·19-s + 3.05·21-s − 5.73·23-s + 25-s − 8.29·27-s − 6.92·29-s + 4.47·31-s − 1.02·35-s + 4.78·37-s − 1.49·39-s − 11.8·41-s + 7.10·43-s + 5.79·45-s − 0.182·47-s − 5.94·49-s − 11.6·51-s + 12.4·53-s + 7.42·57-s − 10.5·59-s + 10.9·61-s + ⋯
L(s)  = 1  − 1.71·3-s + 0.447·5-s − 0.388·7-s + 1.93·9-s + 0.139·13-s − 0.765·15-s + 0.953·17-s − 0.574·19-s + 0.665·21-s − 1.19·23-s + 0.200·25-s − 1.59·27-s − 1.28·29-s + 0.804·31-s − 0.173·35-s + 0.786·37-s − 0.239·39-s − 1.84·41-s + 1.08·43-s + 0.864·45-s − 0.0266·47-s − 0.848·49-s − 1.63·51-s + 1.71·53-s + 0.983·57-s − 1.37·59-s + 1.39·61-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 \( 1 + 2.96T + 3T^{2} \)
7 \( 1 + 1.02T + 7T^{2} \)
13 \( 1 - 0.504T + 13T^{2} \)
17 \( 1 - 3.93T + 17T^{2} \)
19 \( 1 + 2.50T + 19T^{2} \)
23 \( 1 + 5.73T + 23T^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 - 4.47T + 31T^{2} \)
37 \( 1 - 4.78T + 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 7.10T + 43T^{2} \)
47 \( 1 + 0.182T + 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 9.76T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 - 7.32T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 4.33T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 + 7.80T + 97T^{2} \)
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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

−7.72125664405812884844855176481, −6.96460625047188400819141370397, −6.22397639973494794760563929532, −5.83957050369178062465101398347, −5.18584766556543072331711453674, −4.35028037059269154736930087422, −3.52062240393997614157804764761, −2.17253712569091146842021731432, −1.12300771710951077959255575604, 0, 1.12300771710951077959255575604, 2.17253712569091146842021731432, 3.52062240393997614157804764761, 4.35028037059269154736930087422, 5.18584766556543072331711453674, 5.83957050369178062465101398347, 6.22397639973494794760563929532, 6.96460625047188400819141370397, 7.72125664405812884844855176481

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