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Properties

Label 2-882-63.38-c1-0-24
Degree $2$
Conductor $882$
Sign $0.537 - 0.843i$
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  + i·2-s + (0.167 + 1.72i)3-s − 4-s + (1.17 − 2.03i)5-s + (−1.72 + 0.167i)6-s − i·8-s + (−2.94 + 0.577i)9-s + (2.03 + 1.17i)10-s + (4.91 − 2.83i)11-s + (−0.167 − 1.72i)12-s + (1.48 − 0.859i)13-s + (3.70 + 1.68i)15-s + 16-s + (0.884 − 1.53i)17-s + (−0.577 − 2.94i)18-s + (−0.986 + 0.569i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.0967 + 0.995i)3-s − 0.5·4-s + (0.525 − 0.909i)5-s + (−0.703 + 0.0684i)6-s − 0.353i·8-s + (−0.981 + 0.192i)9-s + (0.643 + 0.371i)10-s + (1.48 − 0.855i)11-s + (−0.0483 − 0.497i)12-s + (0.413 − 0.238i)13-s + (0.956 + 0.434i)15-s + 0.250·16-s + (0.214 − 0.371i)17-s + (−0.136 − 0.693i)18-s + (−0.226 + 0.130i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.537 - 0.843i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.537 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57746 + 0.864952i\)
\(L(\frac12)\) \(\approx\) \(1.57746 + 0.864952i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.167 - 1.72i)T \)
7 \( 1 \)
good5 \( 1 + (-1.17 + 2.03i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.91 + 2.83i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.48 + 0.859i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.884 + 1.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.986 - 0.569i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.18 - 1.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.59 - 2.07i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.37iT - 31T^{2} \)
37 \( 1 + (-4.59 - 7.96i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.99 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.76 + 3.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.22T + 59T^{2} \)
61 \( 1 + 8.99iT - 61T^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 - 4.52iT - 71T^{2} \)
73 \( 1 + (4.62 + 2.67i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + (6.27 - 10.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.580 - 1.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.97 + 2.29i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.799753727488895297937640540814, −9.420768957461835555031150893029, −8.654657754579007803182048359344, −8.068773119241022194696711858770, −6.55571227769099246870384696805, −5.86749146887827766185791265726, −5.01575073364064455790947853603, −4.17610074522136427131254906615, −3.16702400157684851006006743730, −1.10172082220649300624825526206, 1.28489043551998518625079248510, 2.24954325368431915653955865672, 3.26826271802177039460889301564, 4.42925996357529978071130795599, 5.89510767545108219487302229869, 6.64610352829319502733455885216, 7.23440156224538316297096310602, 8.506671132574983617873143053831, 9.173893598711238198660765094637, 10.09204580222259689350220153749

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