Properties

Label 2-7200-1.1-c1-0-67
Degree $2$
Conductor $7200$
Sign $-1$
Analytic cond. $57.4922$
Root an. cond. $7.58236$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.763·7-s − 5.70·23-s + 6·29-s + 4.47·41-s − 11.2·43-s + 13.7·47-s − 6.41·49-s − 13.4·61-s + 8.18·67-s − 17.7·83-s − 6·89-s + 18·101-s − 20.1·103-s − 6.29·107-s + 13.4·109-s + ⋯
L(s)  = 1  − 0.288·7-s − 1.19·23-s + 1.11·29-s + 0.698·41-s − 1.71·43-s + 1.99·47-s − 0.916·49-s − 1.71·61-s + 0.999·67-s − 1.94·83-s − 0.635·89-s + 1.79·101-s − 1.98·103-s − 0.608·107-s + 1.28·109-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: no
Arithmetic: yes
Character: $\chi_{7200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ -1)\)

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 \( 1 + 0.763T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 5.70T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 - 13.7T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 - 8.18T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 17.7T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−7.62081365392883341695482804143, −6.82036308005866578813864802442, −6.21087200142691704897560517434, −5.54033214653268332821198739754, −4.66346077894810481427661498659, −3.98873067187532353551688351596, −3.12556403686703036773009843628, −2.31924727511527629839171557918, −1.27936797021892842346315534408, 0, 1.27936797021892842346315534408, 2.31924727511527629839171557918, 3.12556403686703036773009843628, 3.98873067187532353551688351596, 4.66346077894810481427661498659, 5.54033214653268332821198739754, 6.21087200142691704897560517434, 6.82036308005866578813864802442, 7.62081365392883341695482804143

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