↑

Properties

Label 2-825-5.4-c3-0-34
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5.42i·2-s + 3i·3-s − 21.4·4-s − 16.2·6-s − 7.69i·7-s − 72.8i·8-s − 9·9-s − 11·11-s − 64.2i·12-s − 24.8i·13-s + 41.7·14-s + 223.·16-s − 15.9i·17-s − 48.8i·18-s − 15.1·19-s + ⋯
L(s)  = 1  + 1.91i·2-s + 0.577i·3-s − 2.67·4-s − 1.10·6-s − 0.415i·7-s − 3.21i·8-s − 0.333·9-s − 0.301·11-s − 1.54i·12-s − 0.530i·13-s + 0.797·14-s + 3.49·16-s − 0.227i·17-s − 0.639i·18-s − 0.182·19-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

\(L(2)\) \(\approx\) \(1.337853987\)
\(L(\frac12)\) \(\approx\) \(1.337853987\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.42iT - 8T^{2} \)
7 \( 1 + 7.69iT - 343T^{2} \)
13 \( 1 + 24.8iT - 2.19e3T^{2} \)
17 \( 1 + 15.9iT - 4.91e3T^{2} \)
19 \( 1 + 15.1T + 6.85e3T^{2} \)
23 \( 1 + 17.7iT - 1.21e4T^{2} \)
29 \( 1 - 128.T + 2.43e4T^{2} \)
31 \( 1 - 219.T + 2.97e4T^{2} \)
37 \( 1 - 92.0iT - 5.06e4T^{2} \)
41 \( 1 + 459.T + 6.89e4T^{2} \)
43 \( 1 + 64.9iT - 7.95e4T^{2} \)
47 \( 1 - 497. iT - 1.03e5T^{2} \)
53 \( 1 - 526. iT - 1.48e5T^{2} \)
59 \( 1 - 578.T + 2.05e5T^{2} \)
61 \( 1 + 221.T + 2.26e5T^{2} \)
67 \( 1 + 860. iT - 3.00e5T^{2} \)
71 \( 1 - 580.T + 3.57e5T^{2} \)
73 \( 1 + 510. iT - 3.89e5T^{2} \)
79 \( 1 + 1.03e3T + 4.93e5T^{2} \)
83 \( 1 + 606. iT - 5.71e5T^{2} \)
89 \( 1 - 23.4T + 7.04e5T^{2} \)
97 \( 1 - 719. iT - 9.12e5T^{2} \)
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

−9.991647634747738763921819105435, −9.047320927251099428769858530705, −8.301052900714397820932798884604, −7.63903198335863928305364168472, −6.69026628224795576758126242600, −5.94489986311442455734048861629, −4.96587645635245544888778507983, −4.38865190526260566825593154124, −3.19057215278539379595302908791, −0.65509718880976434817584810235, 0.63986573779988583038275854889, 1.81393487503975013913538260158, 2.59600989687639132322466372467, 3.62446093411829798716442687730, 4.68908931992931874560240375029, 5.61211940066337639298633050077, 6.91622869438109944327550849099, 8.414798941334635428847592422151, 8.613398160655916159402900325181, 9.860413160806364008269286421179

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