Properties

Label 2-72e2-12.11-c1-0-25
Degree $2$
Conductor $5184$
Sign $1$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.96i·5-s − 7.19·13-s + 7.20i·17-s − 3.80·25-s + 1.27i·29-s + 11.3·37-s + 1.41i·41-s + 7·49-s + 7.07i·53-s + 5.39·61-s + 21.3i·65-s + 13.1·73-s + 21.3·85-s − 5.51i·89-s + 8·97-s + ⋯
L(s)  = 1  − 1.32i·5-s − 1.99·13-s + 1.74i·17-s − 0.760·25-s + 0.236i·29-s + 1.87·37-s + 0.220i·41-s + 49-s + 0.971i·53-s + 0.690·61-s + 2.64i·65-s + 1.54·73-s + 2.32·85-s − 0.584i·89-s + 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (5183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.472664207\)
\(L(\frac12)\) \(\approx\) \(1.472664207\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + 2.96iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 7.19T + 13T^{2} \)
17 \( 1 - 7.20iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 1.27iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 7.07iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 5.39T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 5.51iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−8.113060895490504906690242555203, −7.73358818056306280226128377110, −6.79766761851130566685130877812, −5.90512282240284304788229481496, −5.23515749897040817216075076434, −4.54165886810893450161982865652, −3.98413890076781687611828487419, −2.70807755751182065342865537785, −1.83714056424128204172559856913, −0.76418202422969752001674906828, 0.53854489454888031335303204672, 2.45219871092086943389723068281, 2.52852978264933874074830824163, 3.56514181126161561209769695487, 4.63927313223451585412329548999, 5.21406900061405236171211109271, 6.18843029526990823188166000905, 6.99609708791116622804140114827, 7.32700720501960809393534002482, 7.927538879569691493290280400482

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