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Properties

Label 2-7200-8.5-c1-0-27
Degree $2$
Conductor $7200$
Sign $-0.318 - 0.947i$
Analytic cond. $57.4922$
Root an. cond. $7.58236$
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  + 2.64·7-s + 1.51i·11-s + 3.87i·13-s + 3.31·17-s + 7.08i·19-s − 4.82·23-s − 2.18i·29-s + 7.36·31-s + 7.87i·37-s − 8.72·41-s − 1.01i·43-s + 7.08·47-s − 0.0164·49-s − 4.50i·53-s + 6.79i·59-s + ⋯
L(s)  = 1  + 0.998·7-s + 0.456i·11-s + 1.07i·13-s + 0.803·17-s + 1.62i·19-s − 1.00·23-s − 0.405i·29-s + 1.32·31-s + 1.29i·37-s − 1.36·41-s − 0.155i·43-s + 1.03·47-s − 0.00234·49-s − 0.619i·53-s + 0.885i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.318 - 0.947i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (3601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ -0.318 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.826040266\)
\(L(\frac12)\) \(\approx\) \(1.826040266\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64T + 7T^{2} \)
11 \( 1 - 1.51iT - 11T^{2} \)
13 \( 1 - 3.87iT - 13T^{2} \)
17 \( 1 - 3.31T + 17T^{2} \)
19 \( 1 - 7.08iT - 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 + 2.18iT - 29T^{2} \)
31 \( 1 - 7.36T + 31T^{2} \)
37 \( 1 - 7.87iT - 37T^{2} \)
41 \( 1 + 8.72T + 41T^{2} \)
43 \( 1 + 1.01iT - 43T^{2} \)
47 \( 1 - 7.08T + 47T^{2} \)
53 \( 1 + 4.50iT - 53T^{2} \)
59 \( 1 - 6.79iT - 59T^{2} \)
61 \( 1 - 3.60iT - 61T^{2} \)
67 \( 1 + 1.01iT - 67T^{2} \)
71 \( 1 + 6.72T + 71T^{2} \)
73 \( 1 + 15.5T + 73T^{2} \)
79 \( 1 - 7.36T + 79T^{2} \)
83 \( 1 - 7.74iT - 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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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

−8.228175706656126826551741923009, −7.53018678347145342365293415060, −6.76652945881356568413479479435, −5.99925794489607637760867208770, −5.33792700988885786062575557154, −4.43236553281475482509704380867, −4.03869284504545887022924801940, −2.94631576734249902266597952651, −1.84996894559537639896158414572, −1.35982587631284074765500475673, 0.43972531619804051272388243433, 1.41756880723263118919161951875, 2.52207268992165769782171458240, 3.21658450181436078476813899782, 4.19479741461149903348653337809, 4.95810898867649952698123269149, 5.51733772000854963403339113002, 6.23770020366856999502178130140, 7.16520053658512471131758028473, 7.76997425453736902879983105191

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