↑

Properties

Label 2-882-441.101-c1-0-26
Degree $2$
Conductor $882$
Sign $0.977 + 0.209i$
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  + (−0.781 + 0.623i)2-s + (1.23 − 1.21i)3-s + (0.222 − 0.974i)4-s + (0.00765 − 0.102i)5-s + (−0.208 + 1.71i)6-s + (0.299 + 2.62i)7-s + (0.433 + 0.900i)8-s + (0.0524 − 2.99i)9-s + (0.0576 + 0.0846i)10-s + (−5.12 − 2.01i)11-s + (−0.908 − 1.47i)12-s + (4.20 + 1.65i)13-s + (−1.87 − 1.86i)14-s + (−0.114 − 0.135i)15-s + (−0.900 − 0.433i)16-s + (6.92 + 2.13i)17-s + ⋯
L(s)  = 1  + (−0.552 + 0.440i)2-s + (0.713 − 0.700i)3-s + (0.111 − 0.487i)4-s + (0.00342 − 0.0456i)5-s + (−0.0853 + 0.701i)6-s + (0.113 + 0.993i)7-s + (0.153 + 0.318i)8-s + (0.0174 − 0.999i)9-s + (0.0182 + 0.0267i)10-s + (−1.54 − 0.606i)11-s + (−0.262 − 0.425i)12-s + (1.16 + 0.458i)13-s + (−0.500 − 0.499i)14-s + (−0.0295 − 0.0349i)15-s + (−0.225 − 0.108i)16-s + (1.68 + 0.518i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.977 + 0.209i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.977 + 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55845 - 0.164803i\)
\(L(\frac12)\) \(\approx\) \(1.55845 - 0.164803i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 - 0.623i)T \)
3 \( 1 + (-1.23 + 1.21i)T \)
7 \( 1 + (-0.299 - 2.62i)T \)
good5 \( 1 + (-0.00765 + 0.102i)T + (-4.94 - 0.745i)T^{2} \)
11 \( 1 + (5.12 + 2.01i)T + (8.06 + 7.48i)T^{2} \)
13 \( 1 + (-4.20 - 1.65i)T + (9.52 + 8.84i)T^{2} \)
17 \( 1 + (-6.92 - 2.13i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-0.0178 + 0.0102i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.44 + 5.86i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (-0.735 + 2.38i)T + (-23.9 - 16.3i)T^{2} \)
31 \( 1 - 1.90iT - 31T^{2} \)
37 \( 1 + (1.09 - 1.01i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (-4.75 - 3.24i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-1.56 + 1.06i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (6.30 + 7.90i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-6.36 + 6.86i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-0.462 - 0.222i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (0.463 - 0.105i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + (11.9 + 2.73i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (0.690 - 0.271i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + 5.86T + 79T^{2} \)
83 \( 1 + (-6.59 - 16.7i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (12.2 + 1.84i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + (-9.44 - 5.45i)T + (48.5 + 84.0i)T^{2} \)
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

−9.954760535177589730753428801585, −8.823800945863346919760672226500, −8.459720343755648665114332199481, −7.84796625353647777226465620462, −6.76488680828252782104285760380, −5.93198488962011493078740325395, −5.12956325776239442118473767944, −3.35425323686375645266901081405, −2.45971599131343486691766777358, −1.06193711957575263199545992844, 1.21739066806626280234357648073, 2.86177488500642180757260009156, 3.46631388272416475351524017131, 4.67184848730474573427367226211, 5.56130421274129178715552796678, 7.37542139175767223479511070670, 7.63332777324666037328391070381, 8.549978815152363377926200705579, 9.484707924491262844490403181554, 10.24658523619777850810709754229

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