Properties

Label 2-7200-1.1-c1-0-5
Degree $2$
Conductor $7200$
Sign $1$
Analytic cond. $57.4922$
Root an. cond. $7.58236$
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  − 2·7-s − 6·11-s − 2·13-s + 6·17-s − 4·19-s − 8·23-s + 8·31-s + 2·37-s + 6·41-s − 4·43-s − 4·47-s − 3·49-s − 6·53-s − 6·59-s − 6·61-s − 4·71-s + 12·73-s + 12·77-s + 8·79-s + 12·83-s − 14·89-s + 4·91-s + 8·97-s − 8·101-s + 6·103-s + 12·107-s + 2·109-s + ⋯
L(s)  = 1  − 0.755·7-s − 1.80·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 1.66·23-s + 1.43·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s − 0.824·53-s − 0.781·59-s − 0.768·61-s − 0.474·71-s + 1.40·73-s + 1.36·77-s + 0.900·79-s + 1.31·83-s − 1.48·89-s + 0.419·91-s + 0.812·97-s − 0.796·101-s + 0.591·103-s + 1.16·107-s + 0.191·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9376935455\)
\(L(\frac12)\) \(\approx\) \(0.9376935455\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−8.006568949128154443512142779685, −7.39615714033765278858179141282, −6.34766372361020732496243981821, −5.94386935190665346825958591813, −5.08826515061587255636774819624, −4.44461603354864387465855783576, −3.38439521168140531407913167308, −2.77590220776225724776448447072, −1.96049499748626916377323042553, −0.46157579209639180855107130732, 0.46157579209639180855107130732, 1.96049499748626916377323042553, 2.77590220776225724776448447072, 3.38439521168140531407913167308, 4.44461603354864387465855783576, 5.08826515061587255636774819624, 5.94386935190665346825958591813, 6.34766372361020732496243981821, 7.39615714033765278858179141282, 8.006568949128154443512142779685

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