Properties

Label 2-7200-1.1-c1-0-38
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.23·7-s − 7.70·23-s + 6·29-s − 4.47·41-s + 6.76·43-s − 0.291·47-s + 20.4·49-s + 13.4·61-s + 14.1·67-s + 4.29·83-s − 6·89-s + 18·101-s − 2.18·103-s + 19.7·107-s − 13.4·109-s + ⋯
L(s)  = 1  + 1.97·7-s − 1.60·23-s + 1.11·29-s − 0.698·41-s + 1.03·43-s − 0.0425·47-s + 2.91·49-s + 1.71·61-s + 1.73·67-s + 0.471·83-s − 0.635·89-s + 1.79·101-s − 0.214·103-s + 1.90·107-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: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.706710580\)
\(L(\frac12)\) \(\approx\) \(2.706710580\)
\(L(\frac{3}{2})\) 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 \( 1 - 5.23T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 7.70T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 - 6.76T + 43T^{2} \)
47 \( 1 + 0.291T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4.29T + 83T^{2} \)
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

−8.096559439968840208930334390025, −7.38196287531328426459859164677, −6.55304506873024720074428570896, −5.67834446320787553307869061323, −5.07503917300434549897758671992, −4.40062423072192247189511174106, −3.76033628270095452825522072513, −2.46129744287166072400837008542, −1.84434307512047331684423106027, −0.869168304842140774857568928108, 0.869168304842140774857568928108, 1.84434307512047331684423106027, 2.46129744287166072400837008542, 3.76033628270095452825522072513, 4.40062423072192247189511174106, 5.07503917300434549897758671992, 5.67834446320787553307869061323, 6.55304506873024720074428570896, 7.38196287531328426459859164677, 8.096559439968840208930334390025

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