Properties

Label 2-60e2-1.1-c1-0-21
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 6·11-s + 5·13-s + 6·17-s − 5·19-s − 6·23-s + 6·29-s + 31-s + 2·37-s + 43-s + 6·47-s − 6·49-s + 12·53-s − 6·59-s − 13·61-s − 11·67-s + 2·73-s + 6·77-s − 8·79-s − 6·83-s + 5·91-s − 7·97-s + 12·101-s + 4·103-s + 12·107-s − 7·109-s − 12·113-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.80·11-s + 1.38·13-s + 1.45·17-s − 1.14·19-s − 1.25·23-s + 1.11·29-s + 0.179·31-s + 0.328·37-s + 0.152·43-s + 0.875·47-s − 6/7·49-s + 1.64·53-s − 0.781·59-s − 1.66·61-s − 1.34·67-s + 0.234·73-s + 0.683·77-s − 0.900·79-s − 0.658·83-s + 0.524·91-s − 0.710·97-s + 1.19·101-s + 0.394·103-s + 1.16·107-s − 0.670·109-s − 1.12·113-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: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.506340872\)
\(L(\frac12)\) \(\approx\) \(2.506340872\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−8.557863511057997757970268824560, −7.961542507896363173235765905676, −7.00797354199330012097468426598, −6.11097472346481350589046305204, −5.91097970406424547476593156462, −4.48877495654256184592709120945, −3.99298778613570722908140056723, −3.15734679160146745553465738936, −1.76993182691893440967496910692, −1.03428119739066030819519430786, 1.03428119739066030819519430786, 1.76993182691893440967496910692, 3.15734679160146745553465738936, 3.99298778613570722908140056723, 4.48877495654256184592709120945, 5.91097970406424547476593156462, 6.11097472346481350589046305204, 7.00797354199330012097468426598, 7.961542507896363173235765905676, 8.557863511057997757970268824560

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