Properties

Label 2-7200-1.1-c1-0-28
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  + 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: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.127884251\)
\(L(\frac12)\) \(\approx\) \(2.127884251\)
\(L(\frac{3}{2})\) 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

−7.83635595286014022258095023805, −7.33821538064769685917881881518, −6.45660213987737243046529437775, −5.94103262162492931038179789181, −4.87903609808944360686114794223, −4.58767631266704864393775510878, −3.44818114533106543883266761275, −2.79107918921604666135985678376, −1.75800995098082718900107585275, −0.75755906389362835296778853913, 0.75755906389362835296778853913, 1.75800995098082718900107585275, 2.79107918921604666135985678376, 3.44818114533106543883266761275, 4.58767631266704864393775510878, 4.87903609808944360686114794223, 5.94103262162492931038179789181, 6.45660213987737243046529437775, 7.33821538064769685917881881518, 7.83635595286014022258095023805

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