↑

Properties

Label 2-882-441.142-c1-0-0
Degree $2$
Conductor $882$
Sign $-0.645 - 0.763i$
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  + (0.365 − 0.930i)2-s + (−1.52 + 0.815i)3-s + (−0.733 − 0.680i)4-s + (0.425 − 0.204i)5-s + (0.200 + 1.72i)6-s + (−1.99 − 1.73i)7-s + (−0.900 + 0.433i)8-s + (1.67 − 2.49i)9-s + (−0.0352 − 0.470i)10-s + (−0.450 − 0.564i)11-s + (1.67 + 0.441i)12-s + (−0.342 + 0.872i)13-s + (−2.34 + 1.22i)14-s + (−0.482 + 0.659i)15-s + (0.0747 + 0.997i)16-s + (3.32 − 3.08i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.658i)2-s + (−0.882 + 0.470i)3-s + (−0.366 − 0.340i)4-s + (0.190 − 0.0915i)5-s + (0.0819 + 0.702i)6-s + (−0.755 − 0.655i)7-s + (−0.318 + 0.153i)8-s + (0.556 − 0.830i)9-s + (−0.0111 − 0.148i)10-s + (−0.135 − 0.170i)11-s + (0.483 + 0.127i)12-s + (−0.0949 + 0.241i)13-s + (−0.626 + 0.327i)14-s + (−0.124 + 0.170i)15-s + (0.0186 + 0.249i)16-s + (0.805 − 0.747i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.645 - 0.763i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.645 - 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00607278 + 0.0130794i\)
\(L(\frac12)\) \(\approx\) \(0.00607278 + 0.0130794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.365 + 0.930i)T \)
3 \( 1 + (1.52 - 0.815i)T \)
7 \( 1 + (1.99 + 1.73i)T \)
good5 \( 1 + (-0.425 + 0.204i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (0.450 + 0.564i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.342 - 0.872i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-3.32 + 3.08i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (1.41 - 2.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.89 - 8.30i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (8.75 + 2.70i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-2.38 + 4.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.67 + 2.36i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (7.89 + 5.38i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (7.01 - 4.78i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (2.68 - 6.84i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-1.90 + 0.588i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (6.68 - 4.55i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-9.91 + 9.20i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-0.321 + 0.556i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.53 - 11.1i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-7.30 - 1.10i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (0.814 + 1.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.547 + 1.39i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-4.86 - 12.3i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (0.956 - 1.65i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.41216176908755045551407286359, −9.671565852069801003956697364726, −9.416413804735749818647456016611, −7.80254870202199220502526250989, −6.87240056486033734293951265221, −5.78293690753383755818535219249, −5.25218772737899306521329308667, −3.93518543536385206453732316399, −3.41637785038923079162163374484, −1.58173876925176896794756189256, 0.00697078665027376412772577996, 2.07049272463141309797067980479, 3.48934744129564122450456725912, 4.84447776701530967812822193162, 5.58235491569056925243078975521, 6.43075991424326471305608805307, 6.89060263923793385216804918702, 8.062921950896398220886452809167, 8.761851385508356383921050472669, 10.06345679451404047047889853013

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