Properties

Label 2-6240-1.1-c1-0-2
Degree $2$
Conductor $6240$
Sign $1$
Analytic cond. $49.8266$
Root an. cond. $7.05879$
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  − 3-s − 5-s − 1.35·7-s + 9-s − 3.58·11-s + 13-s + 15-s − 7.81·17-s − 4.94·19-s + 1.35·21-s − 0.0737·23-s + 25-s − 27-s + 5.51·29-s − 4·31-s + 3.58·33-s + 1.35·35-s − 1.58·37-s − 39-s − 5.58·41-s − 7.17·43-s − 45-s − 2.56·47-s − 5.15·49-s + 7.81·51-s + 12.7·53-s + 3.58·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.513·7-s + 0.333·9-s − 1.08·11-s + 0.277·13-s + 0.258·15-s − 1.89·17-s − 1.13·19-s + 0.296·21-s − 0.0153·23-s + 0.200·25-s − 0.192·27-s + 1.02·29-s − 0.718·31-s + 0.624·33-s + 0.229·35-s − 0.261·37-s − 0.160·39-s − 0.872·41-s − 1.09·43-s − 0.149·45-s − 0.374·47-s − 0.736·49-s + 1.09·51-s + 1.75·53-s + 0.483·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6240\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(49.8266\)
Root analytic conductor: \(7.05879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4532106112\)
\(L(\frac12)\) \(\approx\) \(0.4532106112\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 - T \)
good7 \( 1 + 1.35T + 7T^{2} \)
11 \( 1 + 3.58T + 11T^{2} \)
17 \( 1 + 7.81T + 17T^{2} \)
19 \( 1 + 4.94T + 19T^{2} \)
23 \( 1 + 0.0737T + 23T^{2} \)
29 \( 1 - 5.51T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 1.58T + 37T^{2} \)
41 \( 1 + 5.58T + 41T^{2} \)
43 \( 1 + 7.17T + 43T^{2} \)
47 \( 1 + 2.56T + 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 - 8.45T + 59T^{2} \)
61 \( 1 + 4.30T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 0.871T + 71T^{2} \)
73 \( 1 - 6.79T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 - 5.43T + 83T^{2} \)
89 \( 1 + 6.87T + 89T^{2} \)
97 \( 1 - 7.35T + 97T^{2} \)
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

−8.264454503209869781228117862104, −7.03736772442931028999836293926, −6.76394350716210226102843268728, −5.99294247676327140350225776342, −5.11317096441213363646525872820, −4.50141902085772549251461740741, −3.73739995862293131032922008528, −2.73324285196159800349578383176, −1.88980673205517871866056331925, −0.34177022352854767219290519722, 0.34177022352854767219290519722, 1.88980673205517871866056331925, 2.73324285196159800349578383176, 3.73739995862293131032922008528, 4.50141902085772549251461740741, 5.11317096441213363646525872820, 5.99294247676327140350225776342, 6.76394350716210226102843268728, 7.03736772442931028999836293926, 8.264454503209869781228117862104

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