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Properties

Label 2-882-21.2-c2-0-1
Degree $2$
Conductor $882$
Sign $-0.736 - 0.675i$
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 + (1.22 + 0.707i)5-s − 2.82i·8-s + (−0.999 − 1.73i)10-s + (6.12 − 3.53i)11-s − 15·13-s + (−2.00 + 3.46i)16-s + (−9.79 + 5.65i)17-s + (−6.5 + 11.2i)19-s + 2.82i·20-s − 10·22-s + (19.5 + 11.3i)23-s + (−11.5 − 19.9i)25-s + (18.3 + 10.6i)26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.244 + 0.141i)5-s − 0.353i·8-s + (−0.0999 − 0.173i)10-s + (0.556 − 0.321i)11-s − 1.15·13-s + (−0.125 + 0.216i)16-s + (−0.576 + 0.332i)17-s + (−0.342 + 0.592i)19-s + 0.141i·20-s − 0.454·22-s + (0.851 + 0.491i)23-s + (−0.460 − 0.796i)25-s + (0.706 + 0.407i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.675i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.736 - 0.675i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ -0.736 - 0.675i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3214780471\)
\(L(\frac12)\) \(\approx\) \(0.3214780471\)
\(L(2)\) 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 \( 1 + (-1.22 - 0.707i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-6.12 + 3.53i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 15T + 169T^{2} \)
17 \( 1 + (9.79 - 5.65i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (6.5 - 11.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-19.5 - 11.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 22.6iT - 841T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (8.5 - 14.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 80.6iT - 1.68e3T^{2} \)
43 \( 1 + 85T + 1.84e3T^{2} \)
47 \( 1 + (-62.4 - 36.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (29.3 - 16.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (78.3 - 45.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (36 - 62.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (21.5 + 37.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 52.3iT - 5.04e3T^{2} \)
73 \( 1 + (47.5 + 82.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (34.5 - 59.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 60.8iT - 6.88e3T^{2} \)
89 \( 1 + (-117. - 67.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 16T + 9.40e3T^{2} \)
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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

−10.36355589738687653867759529042, −9.345570505749675792293941868327, −8.810130232701370028009440810182, −7.78295959087901310839248683863, −6.96310006811695195403076663439, −6.09713155045370277675731228054, −4.90433779198642984932102237546, −3.76140648489348378440398762519, −2.62442256553562528986016741389, −1.51069089239873645298087727740, 0.12491806155648033746577812532, 1.69375424203999293160966340217, 2.84448165132518018074388291106, 4.44554467072740647459483146958, 5.20577298533688085537559027525, 6.41740064416713287894348508943, 7.04201442819008846832267518063, 7.916505786593262004571458703011, 8.925621242820872120535195665854, 9.509909692395036404914886628201

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