↑

Properties

Label 2-882-7.3-c2-0-8
Degree $2$
Conductor $882$
Sign $0.611 - 0.791i$
Analytic cond. $24.0327$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (0.274 − 0.158i)5-s + 2.82·8-s + (−0.388 − 0.224i)10-s + (−1.41 + 2.44i)11-s − 3.11i·13-s + (−2.00 − 3.46i)16-s + (15.5 + 8.98i)17-s + (−16.2 + 9.37i)19-s + 0.634i·20-s + 4·22-s + (−9.75 − 16.9i)23-s + (−12.4 + 21.5i)25-s + (−3.81 + 2.20i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.0549 − 0.0317i)5-s + 0.353·8-s + (−0.0388 − 0.0224i)10-s + (−0.128 + 0.222i)11-s − 0.239i·13-s + (−0.125 − 0.216i)16-s + (0.915 + 0.528i)17-s + (−0.854 + 0.493i)19-s + 0.0317i·20-s + 0.181·22-s + (−0.424 − 0.734i)23-s + (−0.497 + 0.862i)25-s + (−0.146 + 0.0847i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.611 - 0.791i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ 0.611 - 0.791i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.048288780\)
\(L(\frac12)\) \(\approx\) \(1.048288780\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.274 + 0.158i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (1.41 - 2.44i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 3.11iT - 169T^{2} \)
17 \( 1 + (-15.5 - 8.98i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (16.2 - 9.37i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (9.75 + 16.9i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 43.5T + 841T^{2} \)
31 \( 1 + (31.6 + 18.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-6.84 - 11.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 53.9iT - 1.68e3T^{2} \)
43 \( 1 - 7.59T + 1.84e3T^{2} \)
47 \( 1 + (49.1 - 28.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (10.1 - 17.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-77.5 - 44.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (64.5 - 37.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 131.T + 5.04e3T^{2} \)
73 \( 1 + (29.3 + 16.9i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-37.7 - 65.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 131. iT - 6.88e3T^{2} \)
89 \( 1 + (96.3 - 55.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 148. iT - 9.40e3T^{2} \)
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

−10.08947531015588057645781710205, −9.422514339963709961547711667200, −8.302675330823889919187853138894, −7.87174613565276665961873623169, −6.64950775479318861808318300467, −5.68738509434781360721019548194, −4.53537747201537005296833871988, −3.56932321492526215761379262418, −2.42861015786625344043070227909, −1.21668513187747442711662329696, 0.42194871520710560807327120597, 1.99623019010606493011707723111, 3.42702022993811174862126467234, 4.63779072716741593385224686496, 5.55281416331347725869367359786, 6.44791942151088456955454422399, 7.26878305493197607488928747474, 8.150055010134418679605912500246, 8.857477195098847817432093408361, 9.781559273077419175486068214145

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