Properties

Label 2-800-5.4-c3-0-50
Degree $2$
Conductor $800$
Sign $-0.894 - 0.447i$
Analytic cond. $47.2015$
Root an. cond. $6.87033$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.47i·3-s − 31.3i·7-s + 6.99·9-s − 8.94·11-s + 62i·13-s − 46i·17-s − 107.·19-s − 140·21-s − 192. i·23-s − 152. i·27-s + 90·29-s − 152.·31-s + 40.0i·33-s − 214i·37-s + 277.·39-s + ⋯
L(s)  = 1  − 0.860i·3-s − 1.69i·7-s + 0.259·9-s − 0.245·11-s + 1.32i·13-s − 0.656i·17-s − 1.29·19-s − 1.45·21-s − 1.74i·23-s − 1.08i·27-s + 0.576·29-s − 0.880·31-s + 0.211i·33-s − 0.950i·37-s + 1.13·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(47.2015\)
Root analytic conductor: \(6.87033\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9737835207\)
\(L(\frac12)\) \(\approx\) \(0.9737835207\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 4.47iT - 27T^{2} \)
7 \( 1 + 31.3iT - 343T^{2} \)
11 \( 1 + 8.94T + 1.33e3T^{2} \)
13 \( 1 - 62iT - 2.19e3T^{2} \)
17 \( 1 + 46iT - 4.91e3T^{2} \)
19 \( 1 + 107.T + 6.85e3T^{2} \)
23 \( 1 + 192. iT - 1.21e4T^{2} \)
29 \( 1 - 90T + 2.43e4T^{2} \)
31 \( 1 + 152.T + 2.97e4T^{2} \)
37 \( 1 + 214iT - 5.06e4T^{2} \)
41 \( 1 + 10T + 6.89e4T^{2} \)
43 \( 1 - 67.0iT - 7.95e4T^{2} \)
47 \( 1 - 398. iT - 1.03e5T^{2} \)
53 \( 1 - 678iT - 1.48e5T^{2} \)
59 \( 1 - 411.T + 2.05e5T^{2} \)
61 \( 1 - 250T + 2.26e5T^{2} \)
67 \( 1 - 49.1iT - 3.00e5T^{2} \)
71 \( 1 + 366.T + 3.57e5T^{2} \)
73 \( 1 + 522iT - 3.89e5T^{2} \)
79 \( 1 + 876.T + 4.93e5T^{2} \)
83 \( 1 + 380. iT - 5.71e5T^{2} \)
89 \( 1 + 970T + 7.04e5T^{2} \)
97 \( 1 + 934iT - 9.12e5T^{2} \)
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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

−9.394392308519436368264363520578, −8.380547769992236014226950755942, −7.40568510119078044886644859483, −6.92841329893006576580853625040, −6.24934413170939122678204713825, −4.50594532920795169653809638031, −4.14150390545467274529921612365, −2.46713464785166697088206785957, −1.33194586535707554332950589761, −0.25815319372157042234637789427, 1.80087276109114388990115333621, 2.98117426256350512159249578057, 3.95403084669979394036208221374, 5.28866899065239042374742781497, 5.57544084503013976553131607298, 6.81105134158700328958139090075, 8.149248720259187538052384987214, 8.644854675887144877967830322096, 9.630085796624385972814856461537, 10.21648314435144520079304243570

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