↑

Properties

Label 2-882-63.38-c1-0-23
Degree $2$
Conductor $882$
Sign $0.569 + 0.821i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (−0.765 − 1.55i)3-s − 4-s + (−0.474 + 0.821i)5-s + (1.55 − 0.765i)6-s − i·8-s + (−1.82 + 2.37i)9-s + (−0.821 − 0.474i)10-s + (−1.51 + 0.876i)11-s + (0.765 + 1.55i)12-s + (0.720 − 0.416i)13-s + (1.64 + 0.108i)15-s + 16-s + (2.73 − 4.73i)17-s + (−2.37 − 1.82i)18-s + (3.21 − 1.85i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.441 − 0.897i)3-s − 0.5·4-s + (−0.212 + 0.367i)5-s + (0.634 − 0.312i)6-s − 0.353i·8-s + (−0.609 + 0.792i)9-s + (−0.259 − 0.150i)10-s + (−0.457 + 0.264i)11-s + (0.220 + 0.448i)12-s + (0.199 − 0.115i)13-s + (0.423 + 0.0280i)15-s + 0.250·16-s + (0.663 − 1.14i)17-s + (−0.560 − 0.431i)18-s + (0.736 − 0.425i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.827043 - 0.433135i\)
\(L(\frac12)\) \(\approx\) \(0.827043 - 0.433135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.765 + 1.55i)T \)
7 \( 1 \)
good5 \( 1 + (0.474 - 0.821i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.51 - 0.876i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.720 + 0.416i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.73 + 4.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.21 + 1.85i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.888 - 0.513i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.58 + 3.80i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 11.0iT - 31T^{2} \)
37 \( 1 + (2.19 + 3.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.05 + 7.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.08 + 5.34i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.80T + 47T^{2} \)
53 \( 1 + (-1.88 - 1.08i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + 1.09iT - 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 + 11.0iT - 71T^{2} \)
73 \( 1 + (3.16 + 1.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 9.84T + 79T^{2} \)
83 \( 1 + (2.41 - 4.17i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.15 - 3.74i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.90 + 2.25i)T + (48.5 + 84.0i)T^{2} \)
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

−9.886849443020991188398441395955, −9.068939710794885484597314749197, −7.86085061420820530511739274168, −7.44340288081499179658067718134, −6.76516511665313501826445943812, −5.61476433391505415882671997289, −5.15177978616590920221442358315, −3.62660875857034468366093385069, −2.31007304217632136780653232126, −0.53633316413634416935545101922, 1.27312316745455171088028160419, 3.10080270282477412305021659537, 3.81244570375924873439441435428, 4.92474606633627251037988661848, 5.53392009563744699873022669162, 6.67470293551033336995210566703, 8.153994356233221753307206330535, 8.670692591390358543946982239366, 9.706915223742583873610089516561, 10.30681531187440103565437380417

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