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Properties

Label 2-882-147.104-c1-0-4
Degree $2$
Conductor $882$
Sign $-0.0734 - 0.997i$
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  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (1.35 + 0.651i)5-s + (1.05 + 2.42i)7-s + (−0.433 + 0.900i)8-s + (0.651 + 1.35i)10-s + (1.68 + 1.34i)11-s + (−2.26 − 1.80i)13-s + (−0.691 + 2.55i)14-s + (−0.900 + 0.433i)16-s + (−0.0455 + 0.199i)17-s + 5.39i·19-s + (−0.334 + 1.46i)20-s + (0.480 + 2.10i)22-s + (−1.31 + 0.300i)23-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (0.111 + 0.487i)4-s + (0.604 + 0.291i)5-s + (0.397 + 0.917i)7-s + (−0.153 + 0.318i)8-s + (0.206 + 0.427i)10-s + (0.508 + 0.405i)11-s + (−0.628 − 0.501i)13-s + (−0.184 + 0.682i)14-s + (−0.225 + 0.108i)16-s + (−0.0110 + 0.0483i)17-s + 1.23i·19-s + (−0.0747 + 0.327i)20-s + (0.102 + 0.448i)22-s + (−0.274 + 0.0626i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0734 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0734 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0734 - 0.997i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.0734 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61348 + 1.73671i\)
\(L(\frac12)\) \(\approx\) \(1.61348 + 1.73671i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 - 0.623i)T \)
3 \( 1 \)
7 \( 1 + (-1.05 - 2.42i)T \)
good5 \( 1 + (-1.35 - 0.651i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-1.68 - 1.34i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (2.26 + 1.80i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.0455 - 0.199i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 5.39iT - 19T^{2} \)
23 \( 1 + (1.31 - 0.300i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-9.29 - 2.12i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 7.31iT - 31T^{2} \)
37 \( 1 + (0.0944 - 0.413i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-1.15 - 0.556i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (8.01 - 3.86i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (2.87 - 3.60i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-7.68 + 1.75i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-5.96 + 2.87i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (1.14 + 0.260i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 1.42T + 67T^{2} \)
71 \( 1 + (-10.7 + 2.44i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (5.67 - 4.52i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + (-4.55 - 5.71i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (4.14 + 5.20i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 18.0iT - 97T^{2} \)
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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

−10.15133270455694844449404688161, −9.645664933907851826663659816800, −8.446513932984298222653708543257, −7.87184248604049994525723118997, −6.68596567442632579470243243888, −5.99226274582739406956770550092, −5.21501625491200967957606852342, −4.22347909277725910323043410744, −2.90010114637551658941791329348, −1.90764359612071326234978132699, 1.01213358866575292616218436375, 2.25552841031723085806084641032, 3.54991839723126251980207537547, 4.60490321583488173640370979550, 5.22881208967227920596421747160, 6.50501020950965967066550473230, 7.08209943712051525570404461267, 8.364160455754511380042077031030, 9.234623480360755874903247074997, 10.07078455669481799256280873684

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