Properties

Label 2-60e2-1.1-c1-0-27
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  − 3·7-s + 2·11-s − 3·13-s − 6·17-s + 7·19-s + 6·23-s + 2·29-s + 5·31-s + 10·37-s − 12·41-s − 3·43-s − 10·47-s + 2·49-s − 6·59-s − 13·61-s − 7·67-s − 4·71-s − 6·73-s − 6·77-s + 8·79-s − 6·83-s − 16·89-s + 9·91-s − 7·97-s + 12·103-s − 16·107-s + 9·109-s + ⋯
L(s)  = 1  − 1.13·7-s + 0.603·11-s − 0.832·13-s − 1.45·17-s + 1.60·19-s + 1.25·23-s + 0.371·29-s + 0.898·31-s + 1.64·37-s − 1.87·41-s − 0.457·43-s − 1.45·47-s + 2/7·49-s − 0.781·59-s − 1.66·61-s − 0.855·67-s − 0.474·71-s − 0.702·73-s − 0.683·77-s + 0.900·79-s − 0.658·83-s − 1.69·89-s + 0.943·91-s − 0.710·97-s + 1.18·103-s − 1.54·107-s + 0.862·109-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: $\chi_{3600} (1, \cdot )$
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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−8.217130195737089314643827465817, −7.25402032673487199321934238648, −6.71649189153655010536111525845, −6.13603427500219351094103491808, −5.02564735992692304106368945332, −4.43513048617502010957575901259, −3.22800736564922508959940330825, −2.79002466592839737156701580419, −1.37918655758222598324442176639, 0, 1.37918655758222598324442176639, 2.79002466592839737156701580419, 3.22800736564922508959940330825, 4.43513048617502010957575901259, 5.02564735992692304106368945332, 6.13603427500219351094103491808, 6.71649189153655010536111525845, 7.25402032673487199321934238648, 8.217130195737089314643827465817

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