Properties

Label 2-4840-1.1-c1-0-34
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.32·3-s + 5-s + 4.78·7-s + 2.40·9-s − 0.601·13-s − 2.32·15-s − 2.64·17-s + 2.60·19-s − 11.1·21-s + 5.17·23-s + 25-s + 1.38·27-s + 5.44·29-s − 10.4·31-s + 4.78·35-s + 9.39·37-s + 1.39·39-s − 4.54·41-s − 3.48·43-s + 2.40·45-s + 5.07·47-s + 15.9·49-s + 6.15·51-s + 3.76·53-s − 6.04·57-s + 11.7·59-s + 12.6·61-s + ⋯
L(s)  = 1  − 1.34·3-s + 0.447·5-s + 1.80·7-s + 0.801·9-s − 0.166·13-s − 0.600·15-s − 0.642·17-s + 0.596·19-s − 2.42·21-s + 1.07·23-s + 0.200·25-s + 0.266·27-s + 1.01·29-s − 1.87·31-s + 0.809·35-s + 1.54·37-s + 0.223·39-s − 0.710·41-s − 0.531·43-s + 0.358·45-s + 0.740·47-s + 2.27·49-s + 0.862·51-s + 0.517·53-s − 0.801·57-s + 1.53·59-s + 1.61·61-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

\(L(1)\) \(\approx\) \(1.678827711\)
\(L(\frac12)\) \(\approx\) \(1.678827711\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.32T + 3T^{2} \)
7 \( 1 - 4.78T + 7T^{2} \)
13 \( 1 + 0.601T + 13T^{2} \)
17 \( 1 + 2.64T + 17T^{2} \)
19 \( 1 - 2.60T + 19T^{2} \)
23 \( 1 - 5.17T + 23T^{2} \)
29 \( 1 - 5.44T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 - 9.39T + 37T^{2} \)
41 \( 1 + 4.54T + 41T^{2} \)
43 \( 1 + 3.48T + 43T^{2} \)
47 \( 1 - 5.07T + 47T^{2} \)
53 \( 1 - 3.76T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 + 15.0T + 67T^{2} \)
71 \( 1 - 0.346T + 71T^{2} \)
73 \( 1 - 3.68T + 73T^{2} \)
79 \( 1 + 8.05T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 - 0.462T + 89T^{2} \)
97 \( 1 - 7.97T + 97T^{2} \)
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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

−8.330334470198394461013134919541, −7.31804299957349209326078045869, −6.89171087096223442330712643477, −5.84201025081897834015901059030, −5.33643936231355906727038274196, −4.84004779841242537761961877474, −4.12574007100460366029484904264, −2.67387234298055596892021370119, −1.65940070019316151238944338334, −0.817225901353903119133286008740, 0.817225901353903119133286008740, 1.65940070019316151238944338334, 2.67387234298055596892021370119, 4.12574007100460366029484904264, 4.84004779841242537761961877474, 5.33643936231355906727038274196, 5.84201025081897834015901059030, 6.89171087096223442330712643477, 7.31804299957349209326078045869, 8.330334470198394461013134919541

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