↑

Properties

Label 2-825-5.4-c3-0-74
Degree $2$
Conductor $825$
Sign $-0.447 + 0.894i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.56i·2-s + 3i·3-s + 5.56·4-s + 4.68·6-s − 16.7i·7-s − 21.1i·8-s − 9·9-s − 11·11-s + 16.6i·12-s + 61.6i·13-s − 26.0·14-s + 11.4·16-s − 81.9i·17-s + 14.0i·18-s + 66.0·19-s + ⋯
L(s)  = 1  − 0.552i·2-s + 0.577i·3-s + 0.695·4-s + 0.318·6-s − 0.902i·7-s − 0.935i·8-s − 0.333·9-s − 0.301·11-s + 0.401i·12-s + 1.31i·13-s − 0.497·14-s + 0.178·16-s − 1.16i·17-s + 0.184i·18-s + 0.797·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.447 + 0.894i$
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: \((2,\ 825,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.921398082\)
\(L(\frac12)\) \(\approx\) \(1.921398082\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.56iT - 8T^{2} \)
7 \( 1 + 16.7iT - 343T^{2} \)
13 \( 1 - 61.6iT - 2.19e3T^{2} \)
17 \( 1 + 81.9iT - 4.91e3T^{2} \)
19 \( 1 - 66.0T + 6.85e3T^{2} \)
23 \( 1 + 118. iT - 1.21e4T^{2} \)
29 \( 1 + 4.91T + 2.43e4T^{2} \)
31 \( 1 + 286.T + 2.97e4T^{2} \)
37 \( 1 + 271. iT - 5.06e4T^{2} \)
41 \( 1 - 117.T + 6.89e4T^{2} \)
43 \( 1 - 364. iT - 7.95e4T^{2} \)
47 \( 1 + 465. iT - 1.03e5T^{2} \)
53 \( 1 - 600. iT - 1.48e5T^{2} \)
59 \( 1 + 647.T + 2.05e5T^{2} \)
61 \( 1 + 190.T + 2.26e5T^{2} \)
67 \( 1 + 1.08e3iT - 3.00e5T^{2} \)
71 \( 1 - 811.T + 3.57e5T^{2} \)
73 \( 1 + 1.03e3iT - 3.89e5T^{2} \)
79 \( 1 + 573.T + 4.93e5T^{2} \)
83 \( 1 + 510. iT - 5.71e5T^{2} \)
89 \( 1 - 748.T + 7.04e5T^{2} \)
97 \( 1 + 96.0iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.588451885344418781046378545238, −9.144186861508755252512817385070, −7.61360445288921264808037779960, −7.10570755917786954993104942626, −6.10918312002181588644804141227, −4.83930793833929920184927737946, −3.95962691437352863345851305738, −3.00558906392901431450842667847, −1.86419236062646626328352916158, −0.47651900803829211012722517824, 1.40883952538166597849753858856, 2.49622371052873988242889331148, 3.44587053349331763741365095767, 5.50285162885953905889377247458, 5.55430029997142666132524747627, 6.66311929333797766678044523563, 7.63692012218925510010042904382, 8.090717361766445729347503464009, 9.018679077508671001770234444932, 10.17963558915313279252915420016

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