Properties

Label 2-3120-1.1-c1-0-25
Degree $2$
Conductor $3120$
Sign $1$
Analytic cond. $24.9133$
Root an. cond. $4.99132$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 3·7-s + 9-s + 5·11-s + 13-s − 15-s + 5·17-s − 2·19-s − 3·21-s + 23-s + 25-s − 27-s + 10·29-s + 2·31-s − 5·33-s + 3·35-s − 3·37-s − 39-s − 9·41-s + 4·43-s + 45-s − 10·47-s + 2·49-s − 5·51-s + 9·53-s + 5·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 0.258·15-s + 1.21·17-s − 0.458·19-s − 0.654·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.85·29-s + 0.359·31-s − 0.870·33-s + 0.507·35-s − 0.493·37-s − 0.160·39-s − 1.40·41-s + 0.609·43-s + 0.149·45-s − 1.45·47-s + 2/7·49-s − 0.700·51-s + 1.23·53-s + 0.674·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(24.9133\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.347076694\)
\(L(\frac12)\) \(\approx\) \(2.347076694\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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

−8.582849055534389147053429823048, −8.092828186253584702384612147549, −6.98970109409746550580962933403, −6.45143952249917777900921807109, −5.64070594873197162301522629244, −4.86218754716371112791943230568, −4.18115076091625241953136432080, −3.11202367766983880548407088633, −1.69680021454953043072253508965, −1.09737019560828316914116411893, 1.09737019560828316914116411893, 1.69680021454953043072253508965, 3.11202367766983880548407088633, 4.18115076091625241953136432080, 4.86218754716371112791943230568, 5.64070594873197162301522629244, 6.45143952249917777900921807109, 6.98970109409746550580962933403, 8.092828186253584702384612147549, 8.582849055534389147053429823048

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