Properties

Label 2-882-3.2-c2-0-14
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

\(L(\frac{3}{2})\) \(\approx\) \(1.785453127\)
\(L(\frac12)\) \(\approx\) \(1.785453127\)
\(L(2)\) 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

−9.773172567125420193964807802980, −9.170657549307608869558746810075, −8.197505043876838885171346721139, −7.46638228430010642778196845327, −6.53270812195965430283299446912, −5.85892779683274785563746997203, −4.79013568905701027944009123222, −3.75923726436234364632360657169, −2.60140568911097692391326393597, −0.792999614294037286584586582458, 0.983934921236531911442892866607, 2.10069955843459331158687924849, 3.48434341800496029820808240195, 4.26405032177997103244180749411, 5.36074127679982741914115330568, 6.26656363743895845733921610655, 7.41036329961863486728704744333, 8.573437046482523142514531420503, 8.874836262127587378111643916346, 10.00547031378230015936580832002

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