Properties

Label 2-60e2-1.1-c1-0-14
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  + 4·7-s − 4·11-s + 4·17-s − 4·23-s + 6·29-s − 4·31-s + 8·37-s + 10·41-s + 4·43-s + 4·47-s + 9·49-s − 12·53-s + 4·59-s + 2·61-s − 4·67-s + 8·73-s − 16·77-s + 12·79-s − 4·83-s + 10·89-s − 8·97-s + 2·101-s − 4·103-s − 12·107-s − 2·109-s + 12·113-s + 16·119-s + ⋯
L(s)  = 1  + 1.51·7-s − 1.20·11-s + 0.970·17-s − 0.834·23-s + 1.11·29-s − 0.718·31-s + 1.31·37-s + 1.56·41-s + 0.609·43-s + 0.583·47-s + 9/7·49-s − 1.64·53-s + 0.520·59-s + 0.256·61-s − 0.488·67-s + 0.936·73-s − 1.82·77-s + 1.35·79-s − 0.439·83-s + 1.05·89-s − 0.812·97-s + 0.199·101-s − 0.394·103-s − 1.16·107-s − 0.191·109-s + 1.12·113-s + 1.46·119-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

\(L(1)\) \(\approx\) \(2.197947626\)
\(L(\frac12)\) \(\approx\) \(2.197947626\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.226876458955487965206451000942, −7.919167398419439034107654528927, −7.38800256081297456965591203909, −6.15421204606922070021803661672, −5.46525873318933504623906073832, −4.80344495575906117477491537233, −4.06442993199655403344546353806, −2.85239179698436020117619375903, −2.03268353241543090446682328600, −0.897179834357593733449721741613, 0.897179834357593733449721741613, 2.03268353241543090446682328600, 2.85239179698436020117619375903, 4.06442993199655403344546353806, 4.80344495575906117477491537233, 5.46525873318933504623906073832, 6.15421204606922070021803661672, 7.38800256081297456965591203909, 7.919167398419439034107654528927, 8.226876458955487965206451000942

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