↑

Properties

Label 2-39e2-1.1-c1-0-37
Degree $2$
Conductor $1521$
Sign $-1$
Analytic cond. $12.1452$
Root an. cond. $3.48500$
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  − 2·4-s − 7-s + 4·16-s + 8·19-s − 5·25-s + 2·28-s − 7·31-s − 10·37-s − 13·43-s − 6·49-s − 13·61-s − 8·64-s + 11·67-s + 17·73-s − 16·76-s − 13·79-s + 5·97-s + 10·100-s − 13·103-s − 19·109-s − 4·112-s + ⋯
L(s)  = 1  − 4-s − 0.377·7-s + 16-s + 1.83·19-s − 25-s + 0.377·28-s − 1.25·31-s − 1.64·37-s − 1.98·43-s − 6/7·49-s − 1.66·61-s − 64-s + 1.34·67-s + 1.98·73-s − 1.83·76-s − 1.46·79-s + 0.507·97-s + 100-s − 1.28·103-s − 1.81·109-s − 0.377·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(12.1452\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−9.246648081896017330013249855574, −8.297055012589454449345700381644, −7.58767355032999252048149491025, −6.65868456331485613589822228939, −5.50612702736645726221905314223, −5.05402201819647223478350482271, −3.80935622445626200448953307328, −3.21324983449548768295456518745, −1.54594071954154839824181515422, 0, 1.54594071954154839824181515422, 3.21324983449548768295456518745, 3.80935622445626200448953307328, 5.05402201819647223478350482271, 5.50612702736645726221905314223, 6.65868456331485613589822228939, 7.58767355032999252048149491025, 8.297055012589454449345700381644, 9.246648081896017330013249855574

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