Properties

Label 2-4840-1.1-c1-0-25
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  − 1.73·3-s + 5-s + 0.267·7-s + 3.46·13-s − 1.73·15-s + 2·19-s − 0.464·21-s + 7.46·23-s + 25-s + 5.19·27-s − 4.92·29-s − 1.46·31-s + 0.267·35-s − 4·37-s − 5.99·39-s − 1.92·41-s + 1.73·43-s + 6.66·47-s − 6.92·49-s − 7.46·53-s − 3.46·57-s − 1.46·59-s + 1.53·61-s + 3.46·65-s + 5.73·67-s − 12.9·69-s + ⋯
L(s)  = 1  − 1.00·3-s + 0.447·5-s + 0.101·7-s + 0.960·13-s − 0.447·15-s + 0.458·19-s − 0.101·21-s + 1.55·23-s + 0.200·25-s + 1.00·27-s − 0.915·29-s − 0.262·31-s + 0.0452·35-s − 0.657·37-s − 0.960·39-s − 0.301·41-s + 0.264·43-s + 0.971·47-s − 0.989·49-s − 1.02·53-s − 0.458·57-s − 0.190·59-s + 0.196·61-s + 0.429·65-s + 0.700·67-s − 1.55·69-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

\(L(1)\) \(\approx\) \(1.452260942\)
\(L(\frac12)\) \(\approx\) \(1.452260942\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T + 3T^{2} \)
7 \( 1 - 0.267T + 7T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 7.46T + 23T^{2} \)
29 \( 1 + 4.92T + 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 1.92T + 41T^{2} \)
43 \( 1 - 1.73T + 43T^{2} \)
47 \( 1 - 6.66T + 47T^{2} \)
53 \( 1 + 7.46T + 53T^{2} \)
59 \( 1 + 1.46T + 59T^{2} \)
61 \( 1 - 1.53T + 61T^{2} \)
67 \( 1 - 5.73T + 67T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 - 1.07T + 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + 7.46T + 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−8.334478838327299477908466859843, −7.40282387736873602965041448527, −6.65240158935321571374247941663, −6.07473690634459949053463561798, −5.33647778442636584547480486025, −4.90720525055885104541809783706, −3.74524939751106978373135225266, −2.93133344243853521266691580614, −1.69660833685283916738577004391, −0.72410449708954778097275374831, 0.72410449708954778097275374831, 1.69660833685283916738577004391, 2.93133344243853521266691580614, 3.74524939751106978373135225266, 4.90720525055885104541809783706, 5.33647778442636584547480486025, 6.07473690634459949053463561798, 6.65240158935321571374247941663, 7.40282387736873602965041448527, 8.334478838327299477908466859843

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