Properties

Label 2-882-3.2-c2-0-2
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 − 4i·5-s − 2.82i·8-s + 5.65·10-s − 2.82i·11-s − 12.7·13-s + 4.00·16-s − 4i·17-s − 22.6·19-s + 8.00i·20-s + 4.00·22-s + 36.7i·23-s + 9·25-s − 18i·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s − 0.800i·5-s − 0.353i·8-s + 0.565·10-s − 0.257i·11-s − 0.979·13-s + 0.250·16-s − 0.235i·17-s − 1.19·19-s + 0.400i·20-s + 0.181·22-s + 1.59i·23-s + 0.359·25-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

\(L(\frac{3}{2})\) \(\approx\) \(0.7979295779\)
\(L(\frac12)\) \(\approx\) \(0.7979295779\)
\(L(2)\) 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 \( 1 + 4iT - 25T^{2} \)
11 \( 1 + 2.82iT - 121T^{2} \)
13 \( 1 + 12.7T + 169T^{2} \)
17 \( 1 + 4iT - 289T^{2} \)
19 \( 1 + 22.6T + 361T^{2} \)
23 \( 1 - 36.7iT - 529T^{2} \)
29 \( 1 - 32.5iT - 841T^{2} \)
31 \( 1 - 50.9T + 961T^{2} \)
37 \( 1 + 32T + 1.36e3T^{2} \)
41 \( 1 - 38iT - 1.68e3T^{2} \)
43 \( 1 - 20T + 1.84e3T^{2} \)
47 \( 1 - 20iT - 2.20e3T^{2} \)
53 \( 1 - 94.7iT - 2.80e3T^{2} \)
59 \( 1 + 4iT - 3.48e3T^{2} \)
61 \( 1 + 83.4T + 3.72e3T^{2} \)
67 \( 1 + 48T + 4.48e3T^{2} \)
71 \( 1 - 76.3iT - 5.04e3T^{2} \)
73 \( 1 - 120.T + 5.32e3T^{2} \)
79 \( 1 + 148T + 6.24e3T^{2} \)
83 \( 1 + 80iT - 6.88e3T^{2} \)
89 \( 1 - 106iT - 7.92e3T^{2} \)
97 \( 1 + 154.T + 9.40e3T^{2} \)
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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

−10.05948431790276342903689149872, −9.237522084635056143512448117155, −8.586597518087836546987212597750, −7.73838104132399426682349088712, −6.90820627297090528649355359121, −5.90900750799775535169453624493, −5.00481483976617660077083118511, −4.33275798585194788503316457314, −2.93147448527198877690404276892, −1.28140172616262453753795129192, 0.27229488788256116857157637041, 2.12389168712459658678415954729, 2.82402918858490982687021733281, 4.11187377432161820979494012702, 4.89358714918937044949945249918, 6.25044464049214110665851980855, 6.94270571434340631402145303890, 8.059754946288008646145346550742, 8.819066983848661291342418107018, 9.961454153195400339750903717766

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