Properties

Label 2-4840-1.1-c1-0-41
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  − 0.569·3-s + 5-s + 1.82·7-s − 2.67·9-s + 3.92·13-s − 0.569·15-s + 3.18·17-s + 8.21·19-s − 1.03·21-s + 0.734·23-s + 25-s + 3.23·27-s + 4.42·29-s − 0.396·31-s + 1.82·35-s + 2.99·37-s − 2.23·39-s − 12.0·41-s − 8.72·43-s − 2.67·45-s − 10.5·47-s − 3.68·49-s − 1.81·51-s + 9.84·53-s − 4.67·57-s + 10.0·59-s − 8.50·61-s + ⋯
L(s)  = 1  − 0.328·3-s + 0.447·5-s + 0.688·7-s − 0.891·9-s + 1.08·13-s − 0.147·15-s + 0.772·17-s + 1.88·19-s − 0.226·21-s + 0.153·23-s + 0.200·25-s + 0.622·27-s + 0.822·29-s − 0.0713·31-s + 0.307·35-s + 0.492·37-s − 0.358·39-s − 1.88·41-s − 1.33·43-s − 0.398·45-s − 1.53·47-s − 0.526·49-s − 0.253·51-s + 1.35·53-s − 0.619·57-s + 1.30·59-s − 1.08·61-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

\(L(1)\) \(\approx\) \(2.227245539\)
\(L(\frac12)\) \(\approx\) \(2.227245539\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.569T + 3T^{2} \)
7 \( 1 - 1.82T + 7T^{2} \)
13 \( 1 - 3.92T + 13T^{2} \)
17 \( 1 - 3.18T + 17T^{2} \)
19 \( 1 - 8.21T + 19T^{2} \)
23 \( 1 - 0.734T + 23T^{2} \)
29 \( 1 - 4.42T + 29T^{2} \)
31 \( 1 + 0.396T + 31T^{2} \)
37 \( 1 - 2.99T + 37T^{2} \)
41 \( 1 + 12.0T + 41T^{2} \)
43 \( 1 + 8.72T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 - 9.84T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 8.50T + 61T^{2} \)
67 \( 1 - 1.11T + 67T^{2} \)
71 \( 1 - 5.87T + 71T^{2} \)
73 \( 1 + 4.03T + 73T^{2} \)
79 \( 1 + 3.26T + 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + 6.56T + 89T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−8.352906754201028644487831630820, −7.64483458727256012680995841657, −6.69880305963237720539978609416, −6.04593560382898357742924848404, −5.23872856563774135236646868305, −4.95310161513374810402219269052, −3.53565017663000552729069431882, −3.03430382236065185667963907426, −1.73487436580062157209295784647, −0.892452750775782907678425280926, 0.892452750775782907678425280926, 1.73487436580062157209295784647, 3.03430382236065185667963907426, 3.53565017663000552729069431882, 4.95310161513374810402219269052, 5.23872856563774135236646868305, 6.04593560382898357742924848404, 6.69880305963237720539978609416, 7.64483458727256012680995841657, 8.352906754201028644487831630820

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