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Properties

Label 2-882-63.38-c1-0-6
Degree $2$
Conductor $882$
Sign $-0.267 - 0.963i$
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 + (0.495 + 1.65i)3-s − 4-s + (0.995 − 1.72i)5-s + (1.65 − 0.495i)6-s + i·8-s + (−2.50 + 1.64i)9-s + (−1.72 − 0.995i)10-s + (−5.21 + 3.00i)11-s + (−0.495 − 1.65i)12-s + (−3.43 + 1.98i)13-s + (3.35 + 0.797i)15-s + 16-s + (−0.781 + 1.35i)17-s + (1.64 + 2.50i)18-s + (−4.16 + 2.40i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.285 + 0.958i)3-s − 0.5·4-s + (0.445 − 0.770i)5-s + (0.677 − 0.202i)6-s + 0.353i·8-s + (−0.836 + 0.548i)9-s + (−0.545 − 0.314i)10-s + (−1.57 + 0.907i)11-s + (−0.142 − 0.479i)12-s + (−0.951 + 0.549i)13-s + (0.865 + 0.206i)15-s + 0.250·16-s + (−0.189 + 0.328i)17-s + (0.387 + 0.591i)18-s + (−0.954 + 0.551i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.267 - 0.963i$
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: \((2,\ 882,\ (\ :1/2),\ -0.267 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503708 + 0.662697i\)
\(L(\frac12)\) \(\approx\) \(0.503708 + 0.662697i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.495 - 1.65i)T \)
7 \( 1 \)
good5 \( 1 + (-0.995 + 1.72i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.21 - 3.00i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.43 - 1.98i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.781 - 1.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.16 - 2.40i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.02 - 2.90i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.26 - 3.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.28iT - 31T^{2} \)
37 \( 1 + (3.02 + 5.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.98 + 5.16i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.53 - 7.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.78T + 47T^{2} \)
53 \( 1 + (-7.27 - 4.19i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 1.86iT - 61T^{2} \)
67 \( 1 - 7.59T + 67T^{2} \)
71 \( 1 - 5.48iT - 71T^{2} \)
73 \( 1 + (9.37 + 5.41i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 + (3.35 - 5.81i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.05 + 3.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.1 + 8.14i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.22853157056265959989521950867, −9.754887464422112383100281552242, −8.898131907652560322769385154468, −8.243381363854054859119883417583, −7.11951323324326828291578634399, −5.45372535918043179688379612565, −4.97395245404313466787612528606, −4.18860933282737673295052879443, −2.84144068097674822440698260319, −1.92909712713416157952132134661, 0.35508518950413274081884436382, 2.49512761624631237834145893618, 3.00965246977687883865030296366, 4.88830214177437193277612464117, 5.68894692046931799660693144595, 6.72940809671466473812993466758, 7.08703959458256541255781208988, 8.332132527122243654358994054218, 8.482339662090711390628061202958, 9.961316798874673221538710918380

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