Properties

Label 2-4840-1.1-c1-0-28
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.713·3-s + 5-s + 0.376·7-s − 2.49·9-s − 2.82·13-s + 0.713·15-s − 4.83·17-s + 2.92·19-s + 0.268·21-s + 3.77·23-s + 25-s − 3.91·27-s + 8.44·29-s + 8.04·31-s + 0.376·35-s + 2.83·37-s − 2.01·39-s − 3.72·41-s − 6.48·43-s − 2.49·45-s + 2.58·47-s − 6.85·49-s − 3.45·51-s − 0.0487·53-s + 2.09·57-s − 1.64·59-s + 8.69·61-s + ⋯
L(s)  = 1  + 0.411·3-s + 0.447·5-s + 0.142·7-s − 0.830·9-s − 0.784·13-s + 0.184·15-s − 1.17·17-s + 0.672·19-s + 0.0585·21-s + 0.786·23-s + 0.200·25-s − 0.753·27-s + 1.56·29-s + 1.44·31-s + 0.0635·35-s + 0.466·37-s − 0.323·39-s − 0.581·41-s − 0.988·43-s − 0.371·45-s + 0.377·47-s − 0.979·49-s − 0.483·51-s − 0.00669·53-s + 0.276·57-s − 0.214·59-s + 1.11·61-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

\(L(1)\) \(\approx\) \(2.171794593\)
\(L(\frac12)\) \(\approx\) \(2.171794593\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.713T + 3T^{2} \)
7 \( 1 - 0.376T + 7T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + 4.83T + 17T^{2} \)
19 \( 1 - 2.92T + 19T^{2} \)
23 \( 1 - 3.77T + 23T^{2} \)
29 \( 1 - 8.44T + 29T^{2} \)
31 \( 1 - 8.04T + 31T^{2} \)
37 \( 1 - 2.83T + 37T^{2} \)
41 \( 1 + 3.72T + 41T^{2} \)
43 \( 1 + 6.48T + 43T^{2} \)
47 \( 1 - 2.58T + 47T^{2} \)
53 \( 1 + 0.0487T + 53T^{2} \)
59 \( 1 + 1.64T + 59T^{2} \)
61 \( 1 - 8.69T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 - 0.513T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 2.73T + 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.360058464367880614436114204321, −7.68423323496538761676456214934, −6.65587958861684847977845975050, −6.32855242238158751886934466693, −5.03406011438609915350748349798, −4.89476825474144368764675537941, −3.58775464091254371766961669019, −2.73113120883918772581423138652, −2.17319376204024605895399088886, −0.78048706394468845380077085006, 0.78048706394468845380077085006, 2.17319376204024605895399088886, 2.73113120883918772581423138652, 3.58775464091254371766961669019, 4.89476825474144368764675537941, 5.03406011438609915350748349798, 6.32855242238158751886934466693, 6.65587958861684847977845975050, 7.68423323496538761676456214934, 8.360058464367880614436114204321

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