↑

Properties

Label 2-60e2-1.1-c1-0-37
Degree $2$
Conductor $3600$
Sign $-1$
Analytic cond. $28.7461$
Root an. cond. $5.36154$
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  + 7-s − 7·13-s + 7·19-s − 11·31-s − 10·37-s + 13·43-s − 6·49-s − 61-s − 11·67-s − 10·73-s + 4·79-s − 7·91-s − 19·97-s − 20·103-s + 17·109-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.94·13-s + 1.60·19-s − 1.97·31-s − 1.64·37-s + 1.98·43-s − 6/7·49-s − 0.128·61-s − 1.34·67-s − 1.17·73-s + 0.450·79-s − 0.733·91-s − 1.92·97-s − 1.97·103-s + 1.62·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(28.7461\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
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 \( 1 + T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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

−8.014937735954873453652934654246, −7.34683047172376687636027413408, −7.03561384453469390281478035841, −5.68085035735596359430110830465, −5.22722112033735983896766322272, −4.44225689338308636469328366323, −3.39509130115371609634775252419, −2.51838198165626804258476764128, −1.51104572904162556096400599334, 0, 1.51104572904162556096400599334, 2.51838198165626804258476764128, 3.39509130115371609634775252419, 4.44225689338308636469328366323, 5.22722112033735983896766322272, 5.68085035735596359430110830465, 7.03561384453469390281478035841, 7.34683047172376687636027413408, 8.014937735954873453652934654246

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