Properties

Label 2-6048-1.1-c1-0-33
Degree $2$
Conductor $6048$
Sign $1$
Analytic cond. $48.2935$
Root an. cond. $6.94935$
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  − 5-s + 7-s + 3·11-s + 2·13-s + 2·17-s + 5·19-s + 3·23-s − 4·25-s + 6·29-s + 3·31-s − 35-s − 9·37-s + 3·41-s − 8·43-s + 4·47-s + 49-s + 12·53-s − 3·55-s − 12·59-s + 4·61-s − 2·65-s + 2·67-s − 71-s + 3·77-s + 4·79-s − 6·83-s − 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.904·11-s + 0.554·13-s + 0.485·17-s + 1.14·19-s + 0.625·23-s − 4/5·25-s + 1.11·29-s + 0.538·31-s − 0.169·35-s − 1.47·37-s + 0.468·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 1.64·53-s − 0.404·55-s − 1.56·59-s + 0.512·61-s − 0.248·65-s + 0.244·67-s − 0.118·71-s + 0.341·77-s + 0.450·79-s − 0.658·83-s − 0.216·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(48.2935\)
Root analytic conductor: \(6.94935\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6048,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.271426870\)
\(L(\frac12)\) \(\approx\) \(2.271426870\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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

−8.165337162817333606821916021574, −7.30691855078910226200071670466, −6.79818045825407078335608178338, −5.90523052358193591897133856067, −5.21405749423837514684479542814, −4.37677487707664520459240091880, −3.64190608377348561234163267928, −2.95314183315036339474647477563, −1.66778541810827258749144620151, −0.851869453680430204125556147557, 0.851869453680430204125556147557, 1.66778541810827258749144620151, 2.95314183315036339474647477563, 3.64190608377348561234163267928, 4.37677487707664520459240091880, 5.21405749423837514684479542814, 5.90523052358193591897133856067, 6.79818045825407078335608178338, 7.30691855078910226200071670466, 8.165337162817333606821916021574

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