Properties

Label 2-4840-1.1-c1-0-16
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  − 0.476·3-s − 5-s + 3.34·7-s − 2.77·9-s − 3.82·13-s + 0.476·15-s − 6.59·17-s + 1.82·19-s − 1.59·21-s + 2.77·23-s + 25-s + 2.75·27-s − 0.952·29-s − 4.17·31-s − 3.34·35-s − 2.95·37-s + 1.82·39-s + 4.82·41-s + 9.93·43-s + 2.77·45-s + 3.16·47-s + 4.17·49-s + 3.14·51-s + 7.46·53-s − 0.867·57-s + 2.77·59-s − 11.7·61-s + ⋯
L(s)  = 1  − 0.275·3-s − 0.447·5-s + 1.26·7-s − 0.924·9-s − 1.05·13-s + 0.123·15-s − 1.59·17-s + 0.417·19-s − 0.347·21-s + 0.578·23-s + 0.200·25-s + 0.529·27-s − 0.176·29-s − 0.750·31-s − 0.565·35-s − 0.485·37-s + 0.291·39-s + 0.752·41-s + 1.51·43-s + 0.413·45-s + 0.461·47-s + 0.597·49-s + 0.439·51-s + 1.02·53-s − 0.114·57-s + 0.361·59-s − 1.50·61-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

\(L(1)\) \(\approx\) \(1.292413325\)
\(L(\frac12)\) \(\approx\) \(1.292413325\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.476T + 3T^{2} \)
7 \( 1 - 3.34T + 7T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 + 6.59T + 17T^{2} \)
19 \( 1 - 1.82T + 19T^{2} \)
23 \( 1 - 2.77T + 23T^{2} \)
29 \( 1 + 0.952T + 29T^{2} \)
31 \( 1 + 4.17T + 31T^{2} \)
37 \( 1 + 2.95T + 37T^{2} \)
41 \( 1 - 4.82T + 41T^{2} \)
43 \( 1 - 9.93T + 43T^{2} \)
47 \( 1 - 3.16T + 47T^{2} \)
53 \( 1 - 7.46T + 53T^{2} \)
59 \( 1 - 2.77T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 - 7.42T + 67T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + 7.64T + 73T^{2} \)
79 \( 1 - 8.59T + 79T^{2} \)
83 \( 1 - 4.86T + 83T^{2} \)
89 \( 1 + 1.59T + 89T^{2} \)
97 \( 1 + 3.82T + 97T^{2} \)
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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

−8.265901544680009285737438908917, −7.52260893127013139252070265972, −7.01245568457296308313759998037, −6.00453360109724990704082108768, −5.18825268688341062072016734480, −4.72416591002272410649554759065, −3.90124559686291183831307572176, −2.71209367559071982056435735220, −2.02190864268235215898019130193, −0.61599769355676104377860801447, 0.61599769355676104377860801447, 2.02190864268235215898019130193, 2.71209367559071982056435735220, 3.90124559686291183831307572176, 4.72416591002272410649554759065, 5.18825268688341062072016734480, 6.00453360109724990704082108768, 7.01245568457296308313759998037, 7.52260893127013139252070265972, 8.265901544680009285737438908917

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