Properties

Label 2-882-21.11-c2-0-27
Degree $2$
Conductor $882$
Sign $-0.848 + 0.529i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + (5.25 − 3.03i)5-s − 2.82i·8-s + (4.29 − 7.43i)10-s + (−10.5 − 6.06i)11-s − 18.5·13-s + (−2.00 − 3.46i)16-s + (−9.44 − 5.45i)17-s + (−10 − 17.3i)19-s − 12.1i·20-s − 17.1·22-s + (10.5 − 6.06i)23-s + (5.91 − 10.2i)25-s + (−22.7 + 13.1i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (1.05 − 0.606i)5-s − 0.353i·8-s + (0.429 − 0.743i)10-s + (−0.955 − 0.551i)11-s − 1.42·13-s + (−0.125 − 0.216i)16-s + (−0.555 − 0.320i)17-s + (−0.526 − 0.911i)19-s − 0.606i·20-s − 0.780·22-s + (0.457 − 0.263i)23-s + (0.236 − 0.409i)25-s + (−0.875 + 0.505i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.848 + 0.529i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ -0.848 + 0.529i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.104009950\)
\(L(\frac12)\) \(\approx\) \(2.104009950\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-5.25 + 3.03i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (10.5 + 6.06i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 18.5T + 169T^{2} \)
17 \( 1 + (9.44 + 5.45i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (10 + 17.3i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-10.5 + 6.06i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 41.8iT - 841T^{2} \)
31 \( 1 + (12.5 - 21.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (19 + 32.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 60.6iT - 1.68e3T^{2} \)
43 \( 1 - 83.4T + 1.84e3T^{2} \)
47 \( 1 + (-14.6 + 8.48i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-81.4 - 47.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (50.4 + 29.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (7.83 + 13.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-66.3 + 114. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 12.1iT - 5.04e3T^{2} \)
73 \( 1 + (-38.4 + 66.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (16.8 + 29.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 60.5iT - 6.88e3T^{2} \)
89 \( 1 + (-4.13 + 2.38i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 188.T + 9.40e3T^{2} \)
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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

−9.562507557269818017835657608606, −9.078322792223985114737919694577, −7.86049731435662525500308634372, −6.85366516351134716935724901031, −5.82211168803904416844770694637, −5.12349226072803344236836573408, −4.41264792381700176594970355541, −2.76970534792255958066048598894, −2.15464752480378188193828104304, −0.49413956933431186125404200601, 2.05422216718695377163991415260, 2.69853946963617875354784602318, 4.10410245511532683661150114663, 5.22242256729356398037834559758, 5.77488960316551398961440157072, 6.92537709473381116782140130897, 7.37478281709475090729735393589, 8.535624095263048465644003269674, 9.601542579947874239760839517465, 10.32855756861189028834722742582

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