Properties

Label 2-7200-1.1-c1-0-17
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  + 4·11-s − 2·13-s − 2·17-s − 8·19-s − 4·23-s + 6·29-s − 2·37-s + 6·41-s + 4·43-s + 12·47-s − 7·49-s − 6·53-s + 12·59-s + 14·61-s − 12·67-s − 2·73-s + 8·79-s + 4·83-s − 2·89-s + 14·97-s + 14·101-s + 8·103-s + 12·107-s + 14·109-s + 6·113-s + ⋯
L(s)  = 1  + 1.20·11-s − 0.554·13-s − 0.485·17-s − 1.83·19-s − 0.834·23-s + 1.11·29-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 1.75·47-s − 49-s − 0.824·53-s + 1.56·59-s + 1.79·61-s − 1.46·67-s − 0.234·73-s + 0.900·79-s + 0.439·83-s − 0.211·89-s + 1.42·97-s + 1.39·101-s + 0.788·103-s + 1.16·107-s + 1.34·109-s + 0.564·113-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: $\chi_{7200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.800933544\)
\(L(\frac12)\) \(\approx\) \(1.800933544\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.921476513679381029930942481143, −7.18147055562474599879969961526, −6.37965898072603990612496473676, −6.12426522257465007746358506675, −4.96471063433703571909091386089, −4.26589031535852789505659101099, −3.76818361104577650677990818112, −2.55076714240410676728377427283, −1.91531739888343518258290807796, −0.67022129765624857642580521738, 0.67022129765624857642580521738, 1.91531739888343518258290807796, 2.55076714240410676728377427283, 3.76818361104577650677990818112, 4.26589031535852789505659101099, 4.96471063433703571909091386089, 6.12426522257465007746358506675, 6.37965898072603990612496473676, 7.18147055562474599879969961526, 7.921476513679381029930942481143

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