Properties

Label 2-6024-1.1-c1-0-23
Degree $2$
Conductor $6024$
Sign $1$
Analytic cond. $48.1018$
Root an. cond. $6.93555$
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 − 1.35·5-s − 0.0125·7-s + 9-s − 2.88·11-s + 2.96·13-s − 1.35·15-s − 4.40·17-s − 7.61·19-s − 0.0125·21-s + 4.16·23-s − 3.15·25-s + 27-s + 10.4·29-s + 3.70·31-s − 2.88·33-s + 0.0170·35-s − 2.43·37-s + 2.96·39-s − 3.30·41-s + 12.1·43-s − 1.35·45-s + 3.83·47-s − 6.99·49-s − 4.40·51-s + 0.362·53-s + 3.92·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.607·5-s − 0.00473·7-s + 0.333·9-s − 0.869·11-s + 0.823·13-s − 0.350·15-s − 1.06·17-s − 1.74·19-s − 0.00273·21-s + 0.868·23-s − 0.630·25-s + 0.192·27-s + 1.93·29-s + 0.665·31-s − 0.502·33-s + 0.00288·35-s − 0.399·37-s + 0.475·39-s − 0.515·41-s + 1.85·43-s − 0.202·45-s + 0.558·47-s − 0.999·49-s − 0.616·51-s + 0.0498·53-s + 0.528·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6024\)    =    \(2^{3} \cdot 3 \cdot 251\)
Sign: $1$
Analytic conductor: \(48.1018\)
Root analytic conductor: \(6.93555\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6024,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.810274855\)
\(L(\frac12)\) \(\approx\) \(1.810274855\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
251 \( 1 - T \)
good5 \( 1 + 1.35T + 5T^{2} \)
7 \( 1 + 0.0125T + 7T^{2} \)
11 \( 1 + 2.88T + 11T^{2} \)
13 \( 1 - 2.96T + 13T^{2} \)
17 \( 1 + 4.40T + 17T^{2} \)
19 \( 1 + 7.61T + 19T^{2} \)
23 \( 1 - 4.16T + 23T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
31 \( 1 - 3.70T + 31T^{2} \)
37 \( 1 + 2.43T + 37T^{2} \)
41 \( 1 + 3.30T + 41T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 - 3.83T + 47T^{2} \)
53 \( 1 - 0.362T + 53T^{2} \)
59 \( 1 - 0.961T + 59T^{2} \)
61 \( 1 + 5.32T + 61T^{2} \)
67 \( 1 - 3.82T + 67T^{2} \)
71 \( 1 + 5.91T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 - 7.59T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 - 8.00T + 97T^{2} \)
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

−8.214201265031443330187430267101, −7.52610185467401195795585323102, −6.59812537969009607466514145365, −6.20664921721096099495042517037, −4.95450553462013457809991164754, −4.40288712440680890765820824318, −3.66453859079484277865774406800, −2.74130273621200033117291433320, −2.05648711657190633701066669675, −0.66838583599836567100472141615, 0.66838583599836567100472141615, 2.05648711657190633701066669675, 2.74130273621200033117291433320, 3.66453859079484277865774406800, 4.40288712440680890765820824318, 4.95450553462013457809991164754, 6.20664921721096099495042517037, 6.59812537969009607466514145365, 7.52610185467401195795585323102, 8.214201265031443330187430267101

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