Properties

Label 2-7200-1.1-c1-0-48
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 + 4·11-s + 6·13-s + 2·17-s + 8·19-s − 6·23-s + 2·29-s + 4·31-s − 2·37-s + 10·41-s + 2·43-s − 2·47-s − 3·49-s + 2·53-s + 2·61-s + 6·67-s + 12·71-s − 10·73-s + 8·77-s − 8·79-s − 10·83-s + 6·89-s + 12·91-s − 10·97-s − 14·101-s − 2·103-s − 6·107-s + ⋯
L(s)  = 1  + 0.755·7-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 1.83·19-s − 1.25·23-s + 0.371·29-s + 0.718·31-s − 0.328·37-s + 1.56·41-s + 0.304·43-s − 0.291·47-s − 3/7·49-s + 0.274·53-s + 0.256·61-s + 0.733·67-s + 1.42·71-s − 1.17·73-s + 0.911·77-s − 0.900·79-s − 1.09·83-s + 0.635·89-s + 1.25·91-s − 1.01·97-s − 1.39·101-s − 0.197·103-s − 0.580·107-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

\(L(1)\) \(\approx\) \(3.110318001\)
\(L(\frac12)\) \(\approx\) \(3.110318001\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.064633688004092960117069345459, −7.27020628150024354725777003749, −6.44520694624991371076149064131, −5.86002476437419324780662506219, −5.18813561324691883586212894715, −4.14367779584153192330538372182, −3.74124755537381837342353439654, −2.77755685345806629552167927235, −1.47754397266105599620064410776, −1.06562636146036967890010355554, 1.06562636146036967890010355554, 1.47754397266105599620064410776, 2.77755685345806629552167927235, 3.74124755537381837342353439654, 4.14367779584153192330538372182, 5.18813561324691883586212894715, 5.86002476437419324780662506219, 6.44520694624991371076149064131, 7.27020628150024354725777003749, 8.064633688004092960117069345459

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