Properties

Label 2-882-441.110-c1-0-51
Degree $2$
Conductor $882$
Sign $-0.999 + 0.0203i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.997 − 0.0747i)2-s + (−1.47 − 0.905i)3-s + (0.988 − 0.149i)4-s + (−0.965 − 4.22i)5-s + (−1.54 − 0.792i)6-s + (−1.83 − 1.90i)7-s + (0.974 − 0.222i)8-s + (1.36 + 2.67i)9-s + (−1.27 − 4.14i)10-s + (0.881 − 1.82i)11-s + (−1.59 − 0.674i)12-s + (2.95 − 0.221i)13-s + (−1.96 − 1.76i)14-s + (−2.40 + 7.12i)15-s + (0.955 − 0.294i)16-s + (−3.53 − 0.532i)17-s + ⋯
L(s)  = 1  + (0.705 − 0.0528i)2-s + (−0.852 − 0.522i)3-s + (0.494 − 0.0745i)4-s + (−0.431 − 1.89i)5-s + (−0.628 − 0.323i)6-s + (−0.692 − 0.721i)7-s + (0.344 − 0.0786i)8-s + (0.453 + 0.891i)9-s + (−0.404 − 1.31i)10-s + (0.265 − 0.551i)11-s + (−0.460 − 0.194i)12-s + (0.818 − 0.0613i)13-s + (−0.526 − 0.472i)14-s + (−0.620 + 1.83i)15-s + (0.238 − 0.0736i)16-s + (−0.856 − 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.999 + 0.0203i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.999 + 0.0203i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0125747 - 1.23609i\)
\(L(\frac12)\) \(\approx\) \(0.0125747 - 1.23609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.997 + 0.0747i)T \)
3 \( 1 + (1.47 + 0.905i)T \)
7 \( 1 + (1.83 + 1.90i)T \)
good5 \( 1 + (0.965 + 4.22i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (-0.881 + 1.82i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-2.95 + 0.221i)T + (12.8 - 1.93i)T^{2} \)
17 \( 1 + (3.53 + 0.532i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-2.92 - 1.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.96 + 1.56i)T + (5.11 - 22.4i)T^{2} \)
29 \( 1 + (9.89 - 3.88i)T + (21.2 - 19.7i)T^{2} \)
31 \( 1 + (1.19 + 0.691i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0880 - 0.224i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-2.39 + 2.22i)T + (3.06 - 40.8i)T^{2} \)
43 \( 1 + (-7.00 - 6.49i)T + (3.21 + 42.8i)T^{2} \)
47 \( 1 + (0.504 + 6.73i)T + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (6.40 + 2.51i)T + (38.8 + 36.0i)T^{2} \)
59 \( 1 + (-7.17 - 6.66i)T + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (-0.896 + 5.94i)T + (-58.2 - 17.9i)T^{2} \)
67 \( 1 + (-2.87 + 4.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.84 + 5.45i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (2.28 - 3.34i)T + (-26.6 - 67.9i)T^{2} \)
79 \( 1 + (-4.63 - 8.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.283 + 3.78i)T + (-82.0 - 12.3i)T^{2} \)
89 \( 1 + (-0.863 + 11.5i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (15.8 + 9.13i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.703530022387703597401682115385, −8.845164042523256956042377447147, −7.87858510807689759504493127389, −7.02872861054816897602080350297, −5.98578500210109314709205943219, −5.33888181004620955494278234113, −4.38775037891494754667739941197, −3.63708897229018819383878711234, −1.56372704378261241673414420654, −0.53419487337020186164452822127, 2.37690544132405045501080784363, 3.47139747242093676530234219372, 4.06610294167244555011168327030, 5.49487086375665334682121209797, 6.24600697428860044846416328551, 6.81184373393605190020822261986, 7.55881197852554861862051054930, 9.194212351810576815515495356157, 9.893200506005627495545464130423, 11.00606665002073193861647962608

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