Properties

Label 2-825-5.4-c3-0-55
Degree $2$
Conductor $825$
Sign $0.894 + 0.447i$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
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  i·2-s + 3i·3-s + 7·4-s + 3·6-s − 26i·7-s − 15i·8-s − 9·9-s + 11·11-s + 21i·12-s + 32i·13-s − 26·14-s + 41·16-s + 74i·17-s + 9i·18-s + 60·19-s + ⋯
L(s)  = 1  − 0.353i·2-s + 0.577i·3-s + 0.875·4-s + 0.204·6-s − 1.40i·7-s − 0.662i·8-s − 0.333·9-s + 0.301·11-s + 0.505i·12-s + 0.682i·13-s − 0.496·14-s + 0.640·16-s + 1.05i·17-s + 0.117i·18-s + 0.724·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.791817879\)
\(L(\frac12)\) \(\approx\) \(2.791817879\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
11 \( 1 - 11T \)
good2 \( 1 + iT - 8T^{2} \)
7 \( 1 + 26iT - 343T^{2} \)
13 \( 1 - 32iT - 2.19e3T^{2} \)
17 \( 1 - 74iT - 4.91e3T^{2} \)
19 \( 1 - 60T + 6.85e3T^{2} \)
23 \( 1 - 182iT - 1.21e4T^{2} \)
29 \( 1 - 90T + 2.43e4T^{2} \)
31 \( 1 + 8T + 2.97e4T^{2} \)
37 \( 1 + 66iT - 5.06e4T^{2} \)
41 \( 1 - 422T + 6.89e4T^{2} \)
43 \( 1 + 408iT - 7.95e4T^{2} \)
47 \( 1 + 506iT - 1.03e5T^{2} \)
53 \( 1 + 348iT - 1.48e5T^{2} \)
59 \( 1 - 200T + 2.05e5T^{2} \)
61 \( 1 - 132T + 2.26e5T^{2} \)
67 \( 1 + 1.03e3iT - 3.00e5T^{2} \)
71 \( 1 - 762T + 3.57e5T^{2} \)
73 \( 1 - 542iT - 3.89e5T^{2} \)
79 \( 1 - 550T + 4.93e5T^{2} \)
83 \( 1 - 132iT - 5.71e5T^{2} \)
89 \( 1 + 570T + 7.04e5T^{2} \)
97 \( 1 - 14iT - 9.12e5T^{2} \)
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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

−10.01120333756950394034449735719, −9.198672754388914411758559260146, −7.918501084000657947375197863043, −7.18190199638508235648026802132, −6.41323530984700875373023987972, −5.28928287962058318132939894842, −3.88735800724838438680904365490, −3.60151395813606525078454654794, −2.01719518082683411702678884092, −0.900957363883468340896770641342, 1.01986425254504192699302747427, 2.50121057571767823018062024288, 2.87264938533099546527297128886, 4.81827071383428850865880771738, 5.80459015168028023573784081393, 6.33485141451524277385433051572, 7.31119076985933211315344881999, 8.079264862538169184606742133672, 8.860100217433436023152778046512, 9.796455685942490626700695552492

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