Properties

Label 2-60e2-1.1-c1-0-33
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s + 2·13-s + 2·17-s − 4·19-s + 2·29-s + 10·37-s − 10·41-s + 4·43-s − 8·47-s − 7·49-s − 10·53-s − 4·59-s − 2·61-s + 12·67-s − 8·71-s − 10·73-s − 12·83-s + 6·89-s − 2·97-s − 6·101-s − 16·103-s + 12·107-s + 14·109-s + 2·113-s + ⋯
L(s)  = 1  − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.371·29-s + 1.64·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s − 49-s − 1.37·53-s − 0.520·59-s − 0.256·61-s + 1.46·67-s − 0.949·71-s − 1.17·73-s − 1.31·83-s + 0.635·89-s − 0.203·97-s − 0.597·101-s − 1.57·103-s + 1.16·107-s + 1.34·109-s + 0.188·113-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: $\chi_{3600} (1, \cdot )$
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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.132479360823444390461948599740, −7.62818581639541327480210566204, −6.59119797171035359138537025559, −6.00095562582224088382030415028, −5.10854764303702027975036605471, −4.42469036791545275362320561796, −3.36550561270118716838366308509, −2.59055487455418232206827677965, −1.46780416341541155024498465553, 0, 1.46780416341541155024498465553, 2.59055487455418232206827677965, 3.36550561270118716838366308509, 4.42469036791545275362320561796, 5.10854764303702027975036605471, 6.00095562582224088382030415028, 6.59119797171035359138537025559, 7.62818581639541327480210566204, 8.132479360823444390461948599740

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