Properties

Label 2-60e2-1.1-c1-0-34
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  − 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: \(1\)
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 \( 1 - 6 T + p T^{2} \)
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

−8.314511452179534597820674660221, −7.26879090044825103896297504040, −6.45091253684309488887986189792, −6.43227025058071167883349470739, −4.81921008362917817651377463164, −4.53076729285564080044343456884, −3.45569829363633777629454199150, −2.51682024423688394090859676004, −1.48874975899024327549158439830, 0, 1.48874975899024327549158439830, 2.51682024423688394090859676004, 3.45569829363633777629454199150, 4.53076729285564080044343456884, 4.81921008362917817651377463164, 6.43227025058071167883349470739, 6.45091253684309488887986189792, 7.26879090044825103896297504040, 8.314511452179534597820674660221

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