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Properties

Label 2-882-441.110-c1-0-8
Degree $2$
Conductor $882$
Sign $0.718 - 0.695i$
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 + (−0.829 − 1.52i)3-s + (0.988 − 0.149i)4-s + (0.374 + 1.64i)5-s + (0.940 + 1.45i)6-s + (−2.63 + 0.238i)7-s + (−0.974 + 0.222i)8-s + (−1.62 + 2.52i)9-s + (−0.496 − 1.60i)10-s + (2.20 − 4.57i)11-s + (−1.04 − 1.37i)12-s + (−2.16 + 0.161i)13-s + (2.60 − 0.434i)14-s + (2.18 − 1.93i)15-s + (0.955 − 0.294i)16-s + (−0.395 − 0.0595i)17-s + ⋯
L(s)  = 1  + (−0.705 + 0.0528i)2-s + (−0.478 − 0.877i)3-s + (0.494 − 0.0745i)4-s + (0.167 + 0.733i)5-s + (0.384 + 0.593i)6-s + (−0.995 + 0.0899i)7-s + (−0.344 + 0.0786i)8-s + (−0.541 + 0.840i)9-s + (−0.156 − 0.508i)10-s + (0.664 − 1.37i)11-s + (−0.302 − 0.398i)12-s + (−0.599 + 0.0449i)13-s + (0.697 − 0.116i)14-s + (0.563 − 0.498i)15-s + (0.238 − 0.0736i)16-s + (−0.0958 − 0.0144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.718 - 0.695i$
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: \((2,\ 882,\ (\ :1/2),\ 0.718 - 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.615964 + 0.249429i\)
\(L(\frac12)\) \(\approx\) \(0.615964 + 0.249429i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.829 + 1.52i)T \)
7 \( 1 + (2.63 - 0.238i)T \)
good5 \( 1 + (-0.374 - 1.64i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (-2.20 + 4.57i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (2.16 - 0.161i)T + (12.8 - 1.93i)T^{2} \)
17 \( 1 + (0.395 + 0.0595i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-3.95 - 2.28i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.51 - 4.39i)T + (5.11 - 22.4i)T^{2} \)
29 \( 1 + (1.94 - 0.763i)T + (21.2 - 19.7i)T^{2} \)
31 \( 1 + (-3.15 - 1.82i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.58 - 9.13i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (6.90 - 6.40i)T + (3.06 - 40.8i)T^{2} \)
43 \( 1 + (-6.01 - 5.58i)T + (3.21 + 42.8i)T^{2} \)
47 \( 1 + (0.357 + 4.77i)T + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (-1.66 - 0.654i)T + (38.8 + 36.0i)T^{2} \)
59 \( 1 + (-4.51 - 4.18i)T + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (0.711 - 4.71i)T + (-58.2 - 17.9i)T^{2} \)
67 \( 1 + (-4.33 + 7.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.74 + 7.77i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (2.00 - 2.93i)T + (-26.6 - 67.9i)T^{2} \)
79 \( 1 + (-8.12 - 14.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.686 - 9.16i)T + (-82.0 - 12.3i)T^{2} \)
89 \( 1 + (0.131 - 1.74i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (-2.38 - 1.37i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.14258790884564681097650969525, −9.536642039256099852282528837296, −8.419203686042107178702919722461, −7.67774659659654585472974945235, −6.64843065362552757605663495564, −6.32002458234745527782184476696, −5.41893570058256472227114333120, −3.47129735623016592092252230949, −2.60821621507223236668582328969, −1.09371462278385379156999775397, 0.50467024571617399035963504505, 2.31952822934549109302008748745, 3.76280549882960098048342489552, 4.65674571222238067056049430854, 5.66791551888733839126915474641, 6.65486352629696481721607721430, 7.42660826942492398070800777504, 8.779403875943150991524738177392, 9.404401477564376633380894284822, 9.846828758761172892097551468242

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