↑

Properties

Label 2-882-441.151-c1-0-6
Degree $2$
Conductor $882$
Sign $-0.811 - 0.583i$
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  + (−0.900 + 0.433i)2-s + (1.03 + 1.38i)3-s + (0.623 − 0.781i)4-s + (0.385 + 0.357i)5-s + (−1.53 − 0.802i)6-s + (−1.74 + 1.98i)7-s + (−0.222 + 0.974i)8-s + (−0.859 + 2.87i)9-s + (−0.501 − 0.154i)10-s + (1.08 + 0.737i)11-s + (1.73 + 0.0572i)12-s + (4.58 + 3.12i)13-s + (0.709 − 2.54i)14-s + (−0.0979 + 0.904i)15-s + (−0.222 − 0.974i)16-s + (0.965 − 0.145i)17-s + ⋯
L(s)  = 1  + (−0.637 + 0.306i)2-s + (0.597 + 0.802i)3-s + (0.311 − 0.390i)4-s + (0.172 + 0.159i)5-s + (−0.626 − 0.327i)6-s + (−0.659 + 0.751i)7-s + (−0.0786 + 0.344i)8-s + (−0.286 + 0.958i)9-s + (−0.158 − 0.0489i)10-s + (0.326 + 0.222i)11-s + (0.499 + 0.0165i)12-s + (1.27 + 0.866i)13-s + (0.189 − 0.681i)14-s + (−0.0252 + 0.233i)15-s + (−0.0556 − 0.243i)16-s + (0.234 − 0.0353i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.811 - 0.583i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.811 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.378335 + 1.17443i\)
\(L(\frac12)\) \(\approx\) \(0.378335 + 1.17443i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 - 0.433i)T \)
3 \( 1 + (-1.03 - 1.38i)T \)
7 \( 1 + (1.74 - 1.98i)T \)
good5 \( 1 + (-0.385 - 0.357i)T + (0.373 + 4.98i)T^{2} \)
11 \( 1 + (-1.08 - 0.737i)T + (4.01 + 10.2i)T^{2} \)
13 \( 1 + (-4.58 - 3.12i)T + (4.74 + 12.1i)T^{2} \)
17 \( 1 + (-0.965 + 0.145i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (2.78 + 4.81i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.70 - 6.89i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (6.77 - 1.02i)T + (27.7 - 8.54i)T^{2} \)
31 \( 1 + 0.724T + 31T^{2} \)
37 \( 1 + (0.227 - 0.579i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (10.7 - 3.30i)T + (33.8 - 23.0i)T^{2} \)
43 \( 1 + (-9.86 - 3.04i)T + (35.5 + 24.2i)T^{2} \)
47 \( 1 + (3.22 - 1.55i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (1.23 + 3.15i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (-0.968 - 4.24i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-8.31 - 10.4i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + 8.10T + 67T^{2} \)
71 \( 1 + (0.0628 - 0.0787i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-10.5 + 7.22i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 - 0.726T + 79T^{2} \)
83 \( 1 + (7.53 - 5.13i)T + (30.3 - 77.2i)T^{2} \)
89 \( 1 + (-0.193 - 2.57i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (-0.991 + 1.71i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.22346709334045420093642738922, −9.295412590205541456986624058337, −9.084820242810334173526452069572, −8.244107624521731759462099618912, −7.09017918540269602880178032821, −6.24213768188937553045917596046, −5.32892793686273956734895862294, −4.08193645880749816180293706685, −3.04431290408815786241220254226, −1.85967740199903106875914964049, 0.68474051569016558189423203939, 1.82646891968456485610809169702, 3.27598894901139470409367045745, 3.83503137538623265572130801951, 5.77700047493399604248496174692, 6.54302011399776850026950043806, 7.38433696695677959127690764530, 8.259405712446149058393776534011, 8.829166730689357620132672982051, 9.726238812942400401654867879838

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