Properties

Label 2-882-3.2-c2-0-12
Degree $2$
Conductor $882$
Sign $0.816 + 0.577i$
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  − 1.41i·2-s − 2.00·4-s + 2i·5-s + 2.82i·8-s + 2.82·10-s + 2.82i·11-s − 12.7·13-s + 4.00·16-s − 22i·17-s + 5.65·19-s − 4.00i·20-s + 4.00·22-s − 2.82i·23-s + 21·25-s + 18i·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s + 0.400i·5-s + 0.353i·8-s + 0.282·10-s + 0.257i·11-s − 0.979·13-s + 0.250·16-s − 1.29i·17-s + 0.297·19-s − 0.200i·20-s + 0.181·22-s − 0.122i·23-s + 0.839·25-s + 0.692i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.642741184\)
\(L(\frac12)\) \(\approx\) \(1.642741184\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2iT - 25T^{2} \)
11 \( 1 - 2.82iT - 121T^{2} \)
13 \( 1 + 12.7T + 169T^{2} \)
17 \( 1 + 22iT - 289T^{2} \)
19 \( 1 - 5.65T + 361T^{2} \)
23 \( 1 + 2.82iT - 529T^{2} \)
29 \( 1 - 35.3iT - 841T^{2} \)
31 \( 1 - 33.9T + 961T^{2} \)
37 \( 1 - 64T + 1.36e3T^{2} \)
41 \( 1 - 20iT - 1.68e3T^{2} \)
43 \( 1 - 44T + 1.84e3T^{2} \)
47 \( 1 - 68iT - 2.20e3T^{2} \)
53 \( 1 + 18.3iT - 2.80e3T^{2} \)
59 \( 1 + 100iT - 3.48e3T^{2} \)
61 \( 1 + 52.3T + 3.72e3T^{2} \)
67 \( 1 - 120T + 4.48e3T^{2} \)
71 \( 1 + 8.48iT - 5.04e3T^{2} \)
73 \( 1 - 74.9T + 5.32e3T^{2} \)
79 \( 1 - 92T + 6.24e3T^{2} \)
83 \( 1 - 112iT - 6.88e3T^{2} \)
89 \( 1 + 20iT - 7.92e3T^{2} \)
97 \( 1 - 26.8T + 9.40e3T^{2} \)
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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

−9.656034277269365091970423982460, −9.494817162933210063268102336818, −8.198827491798137733819740041320, −7.34551252542406505638621817932, −6.49135433342551387115967289084, −5.15494788372360179783538372542, −4.51373838564450187056943290538, −3.11662978885447902695817070140, −2.43483088063024155219764390066, −0.847246425636990681161218393380, 0.805067473172015488345116749782, 2.47627188026566385276826774656, 3.92333577198126412740137709391, 4.77635180840864639505962931359, 5.74478191134501118672409423052, 6.51765581789409209756598516428, 7.58099468003946227607638784878, 8.211050586909745868008580443568, 9.078431810470269859053854790284, 9.872764245732024184339860828851

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