Properties

Label 2-882-3.2-c2-0-17
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 − 6.06i·5-s − 2.82i·8-s + 8.58·10-s + 12.1i·11-s + 18.5·13-s + 4.00·16-s − 10.9i·17-s − 20·19-s + 12.1i·20-s − 17.1·22-s + 12.1i·23-s − 11.8·25-s + 26.2i·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s − 1.21i·5-s − 0.353i·8-s + 0.858·10-s + 1.10i·11-s + 1.42·13-s + 0.250·16-s − 0.641i·17-s − 1.05·19-s + 0.606i·20-s − 0.780·22-s + 0.527i·23-s − 0.473·25-s + 1.01i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.597385419\)
\(L(\frac12)\) \(\approx\) \(1.597385419\)
\(L(2)\) 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 \( 1 + 6.06iT - 25T^{2} \)
11 \( 1 - 12.1iT - 121T^{2} \)
13 \( 1 - 18.5T + 169T^{2} \)
17 \( 1 + 10.9iT - 289T^{2} \)
19 \( 1 + 20T + 361T^{2} \)
23 \( 1 - 12.1iT - 529T^{2} \)
29 \( 1 + 41.8iT - 841T^{2} \)
31 \( 1 + 25.1T + 961T^{2} \)
37 \( 1 - 38T + 1.36e3T^{2} \)
41 \( 1 + 60.6iT - 1.68e3T^{2} \)
43 \( 1 - 83.4T + 1.84e3T^{2} \)
47 \( 1 + 16.9iT - 2.20e3T^{2} \)
53 \( 1 + 94.0iT - 2.80e3T^{2} \)
59 \( 1 + 58.2iT - 3.48e3T^{2} \)
61 \( 1 + 15.6T + 3.72e3T^{2} \)
67 \( 1 + 132.T + 4.48e3T^{2} \)
71 \( 1 - 12.1iT - 5.04e3T^{2} \)
73 \( 1 - 76.9T + 5.32e3T^{2} \)
79 \( 1 - 33.6T + 6.24e3T^{2} \)
83 \( 1 + 60.5iT - 6.88e3T^{2} \)
89 \( 1 + 4.77iT - 7.92e3T^{2} \)
97 \( 1 - 188.T + 9.40e3T^{2} \)
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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

−9.484233711258791741243393271729, −8.996836718513461131998685681454, −8.174883950822671422335241177150, −7.39476512965285040014171472071, −6.32478025212860335265152778726, −5.50859034353940230781116776037, −4.55910238973426438417750038752, −3.87090155293225851281059968014, −1.98163545217554938788899857921, −0.60158639314093212695543247492, 1.20517561832429915827037626936, 2.65499430239307134839998459015, 3.43204930669500127619164168794, 4.31819383622217424229097272028, 5.93796801670953561996555737755, 6.31090114922020771723968919292, 7.56192047935155680089750270631, 8.553397975171837808105081769015, 9.132636708222898513875041253091, 10.48843090666776772203703875981

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