Properties

Label 2-882-3.2-c2-0-5
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 + 8.60i·5-s − 2.82i·8-s − 12.1·10-s − 2.82i·11-s + 12.1·13-s + 4.00·16-s + 25.8i·17-s − 24.3·19-s − 17.2i·20-s + 4.00·22-s + 42.4i·23-s − 49·25-s + 17.2i·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s + 1.72i·5-s − 0.353i·8-s − 1.21·10-s − 0.257i·11-s + 0.935·13-s + 0.250·16-s + 1.51i·17-s − 1.28·19-s − 0.860i·20-s + 0.181·22-s + 1.84i·23-s − 1.95·25-s + 0.661i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.115247611\)
\(L(\frac12)\) \(\approx\) \(1.115247611\)
\(L(2)\) 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 \( 1 - 8.60iT - 25T^{2} \)
11 \( 1 + 2.82iT - 121T^{2} \)
13 \( 1 - 12.1T + 169T^{2} \)
17 \( 1 - 25.8iT - 289T^{2} \)
19 \( 1 + 24.3T + 361T^{2} \)
23 \( 1 - 42.4iT - 529T^{2} \)
29 \( 1 + 15.5iT - 841T^{2} \)
31 \( 1 - 24.3T + 961T^{2} \)
37 \( 1 + 6T + 1.36e3T^{2} \)
41 \( 1 - 25.8iT - 1.68e3T^{2} \)
43 \( 1 + 68T + 1.84e3T^{2} \)
47 \( 1 + 68.8iT - 2.20e3T^{2} \)
53 \( 1 + 41.0iT - 2.80e3T^{2} \)
59 \( 1 + 68.8iT - 3.48e3T^{2} \)
61 \( 1 - 97.3T + 3.72e3T^{2} \)
67 \( 1 + 104T + 4.48e3T^{2} \)
71 \( 1 + 70.7iT - 5.04e3T^{2} \)
73 \( 1 - 60.8T + 5.32e3T^{2} \)
79 \( 1 + 20T + 6.24e3T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 - 94.6iT - 7.92e3T^{2} \)
97 \( 1 + 158.T + 9.40e3T^{2} \)
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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

−10.43088091220653480470755424316, −9.716211959685369980897113553953, −8.430741829809953871523840979112, −7.941188950066159964315045683271, −6.72879173463548010364093930558, −6.41453302085304345891122172098, −5.53185322348073154281643353134, −3.94592705394470920443596377872, −3.35788222086694448406450937812, −1.87729059707322405065372134023, 0.37708142159957593234561943418, 1.40581170529869928932968647518, 2.69937931496414362329289544349, 4.22986449759016546900166220730, 4.67546311757176193439419532991, 5.66313341273403205750306222062, 6.82329600200919400285136277497, 8.260871567964461576743393530058, 8.639842530703635643149403561826, 9.355933630483436782333172908302

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