Properties

Label 2-6240-1.1-c1-0-17
Degree $2$
Conductor $6240$
Sign $1$
Analytic cond. $49.8266$
Root an. cond. $7.05879$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 7-s + 9-s + 3·11-s − 13-s − 15-s − 5·17-s + 2·19-s + 21-s + 3·23-s + 25-s − 27-s + 4·31-s − 3·33-s − 35-s + 37-s + 39-s + 9·41-s − 2·43-s + 45-s − 8·47-s − 6·49-s + 5·51-s − 53-s + 3·55-s − 2·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 0.258·15-s − 1.21·17-s + 0.458·19-s + 0.218·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.718·31-s − 0.522·33-s − 0.169·35-s + 0.164·37-s + 0.160·39-s + 1.40·41-s − 0.304·43-s + 0.149·45-s − 1.16·47-s − 6/7·49-s + 0.700·51-s − 0.137·53-s + 0.404·55-s − 0.264·57-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.640360467\)
\(L(\frac12)\) \(\approx\) \(1.640360467\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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

−8.032738011730645770768761631572, −7.10478351307535636375613719077, −6.52872554201688808231341911458, −6.12132471512271349064332017064, −5.13133879437452927879317491456, −4.56802817050921818408484136696, −3.68529075760365404601955128362, −2.73148715599548466722492138220, −1.75712103104138674899765724412, −0.69874911503415318511062226331, 0.69874911503415318511062226331, 1.75712103104138674899765724412, 2.73148715599548466722492138220, 3.68529075760365404601955128362, 4.56802817050921818408484136696, 5.13133879437452927879317491456, 6.12132471512271349064332017064, 6.52872554201688808231341911458, 7.10478351307535636375613719077, 8.032738011730645770768761631572

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