Properties

Label 2-31200-1.1-c1-0-45
Degree $2$
Conductor $31200$
Sign $-1$
Analytic cond. $249.133$
Root an. cond. $15.7839$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·7-s + 9-s + 3·11-s − 13-s + 3·17-s − 2·19-s − 3·21-s − 23-s − 27-s − 8·29-s − 4·31-s − 3·33-s + 5·37-s + 39-s − 7·41-s + 2·43-s − 4·47-s + 2·49-s − 3·51-s + 11·53-s + 2·57-s + 4·59-s + 61-s + 3·63-s + 69-s − 5·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 0.727·17-s − 0.458·19-s − 0.654·21-s − 0.208·23-s − 0.192·27-s − 1.48·29-s − 0.718·31-s − 0.522·33-s + 0.821·37-s + 0.160·39-s − 1.09·41-s + 0.304·43-s − 0.583·47-s + 2/7·49-s − 0.420·51-s + 1.51·53-s + 0.264·57-s + 0.520·59-s + 0.128·61-s + 0.377·63-s + 0.120·69-s − 0.593·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31200\)    =    \(2^{5} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(249.133\)
Root analytic conductor: \(15.7839\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{31200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
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

−15.05083823193082, −14.84690842864557, −14.51351472402466, −13.77726220035413, −13.22533508459770, −12.63319928734002, −11.98733232778293, −11.67603788953134, −11.15571363570863, −10.67894304582174, −10.02740126330525, −9.433810096449105, −8.919122487512332, −8.153208862681646, −7.770033577045982, −7.042667401403306, −6.606696017794163, −5.670086105651670, −5.456210474430585, −4.682231138693020, −4.055259008895575, −3.561706169123540, −2.443819249189099, −1.696666062443609, −1.144422158191774, 0, 1.144422158191774, 1.696666062443609, 2.443819249189099, 3.561706169123540, 4.055259008895575, 4.682231138693020, 5.456210474430585, 5.670086105651670, 6.606696017794163, 7.042667401403306, 7.770033577045982, 8.153208862681646, 8.919122487512332, 9.433810096449105, 10.02740126330525, 10.67894304582174, 11.15571363570863, 11.67603788953134, 11.98733232778293, 12.63319928734002, 13.22533508459770, 13.77726220035413, 14.51351472402466, 14.84690842864557, 15.05083823193082

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