Properties

Label 2-7440-1.1-c1-0-82
Degree $2$
Conductor $7440$
Sign $-1$
Analytic cond. $59.4086$
Root an. cond. $7.70770$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 5.08·7-s + 9-s + 4.17·11-s − 1.08·13-s − 15-s + 0.648·17-s + 2.70·19-s − 5.08·21-s − 7.52·23-s + 25-s + 27-s + 5.90·29-s − 31-s + 4.17·33-s + 5.08·35-s + 0.913·37-s − 1.08·39-s + 2.17·41-s − 8.17·43-s − 45-s + 10.2·47-s + 18.8·49-s + 0.648·51-s + 5.52·53-s − 4.17·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.92·7-s + 0.333·9-s + 1.25·11-s − 0.301·13-s − 0.258·15-s + 0.157·17-s + 0.620·19-s − 1.10·21-s − 1.56·23-s + 0.200·25-s + 0.192·27-s + 1.09·29-s − 0.179·31-s + 0.726·33-s + 0.859·35-s + 0.150·37-s − 0.173·39-s + 0.339·41-s − 1.24·43-s − 0.149·45-s + 1.49·47-s + 2.69·49-s + 0.0907·51-s + 0.758·53-s − 0.562·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7440\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-1$
Analytic conductor: \(59.4086\)
Root analytic conductor: \(7.70770\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7440,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
31 \( 1 + T \)
good7 \( 1 + 5.08T + 7T^{2} \)
11 \( 1 - 4.17T + 11T^{2} \)
13 \( 1 + 1.08T + 13T^{2} \)
17 \( 1 - 0.648T + 17T^{2} \)
19 \( 1 - 2.70T + 19T^{2} \)
23 \( 1 + 7.52T + 23T^{2} \)
29 \( 1 - 5.90T + 29T^{2} \)
37 \( 1 - 0.913T + 37T^{2} \)
41 \( 1 - 2.17T + 41T^{2} \)
43 \( 1 + 8.17T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 - 5.52T + 53T^{2} \)
59 \( 1 + 0.438T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 9.79T + 67T^{2} \)
71 \( 1 + 2.96T + 71T^{2} \)
73 \( 1 + 1.25T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 - 18.1T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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

−7.40778753109173572012781677151, −6.91142475646651734587028612605, −6.26974359568956614732909468392, −5.65831262539133229581415519384, −4.34465530979254364005947437384, −3.86069795207944480913786002575, −3.18867414274944541852562156086, −2.51038764187387803437759237763, −1.21655947750892312251002142585, 0, 1.21655947750892312251002142585, 2.51038764187387803437759237763, 3.18867414274944541852562156086, 3.86069795207944480913786002575, 4.34465530979254364005947437384, 5.65831262539133229581415519384, 6.26974359568956614732909468392, 6.91142475646651734587028612605, 7.40778753109173572012781677151

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