Properties

Label 2-882-147.104-c1-0-12
Degree $2$
Conductor $882$
Sign $0.405 + 0.914i$
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 + (−1.86 − 0.897i)5-s + (2.21 − 1.45i)7-s + (−0.433 + 0.900i)8-s + (−0.897 − 1.86i)10-s + (−4.86 − 3.87i)11-s + (0.0474 + 0.0378i)13-s + (2.63 + 0.244i)14-s + (−0.900 + 0.433i)16-s + (1.28 − 5.61i)17-s − 2.01i·19-s + (0.460 − 2.01i)20-s + (−1.38 − 6.06i)22-s + (−0.713 + 0.162i)23-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (0.111 + 0.487i)4-s + (−0.833 − 0.401i)5-s + (0.836 − 0.548i)7-s + (−0.153 + 0.318i)8-s + (−0.283 − 0.589i)10-s + (−1.46 − 1.16i)11-s + (0.0131 + 0.0105i)13-s + (0.704 + 0.0654i)14-s + (−0.225 + 0.108i)16-s + (0.310 − 1.36i)17-s − 0.462i·19-s + (0.102 − 0.450i)20-s + (−0.295 − 1.29i)22-s + (−0.148 + 0.0339i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.405 + 0.914i$
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: \((2,\ 882,\ (\ :1/2),\ 0.405 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23714 - 0.804756i\)
\(L(\frac12)\) \(\approx\) \(1.23714 - 0.804756i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 1.45i)T \)
good5 \( 1 + (1.86 + 0.897i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (4.86 + 3.87i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.0474 - 0.0378i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.28 + 5.61i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 2.01iT - 19T^{2} \)
23 \( 1 + (0.713 - 0.162i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-8.50 - 1.94i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 0.294iT - 31T^{2} \)
37 \( 1 + (0.570 - 2.49i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (7.61 + 3.66i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (1.03 - 0.500i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-3.82 + 4.79i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (7.87 - 1.79i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-4.79 + 2.30i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-4.45 - 1.01i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 4.72T + 67T^{2} \)
71 \( 1 + (1.80 - 0.412i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-12.0 + 9.60i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 4.58T + 79T^{2} \)
83 \( 1 + (-0.413 - 0.518i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-6.55 - 8.21i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 14.0iT - 97T^{2} \)
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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

−10.16235958867130593297381760786, −8.733133636807891782982201754284, −8.114091753180573174372076024042, −7.58073951153833561633931079649, −6.59771468682184742114793038788, −5.19140761780261637798427285614, −4.92142171389320631898205235285, −3.70944540580090428887100893035, −2.68326212466464719895868074309, −0.58581744733378142996186019809, 1.78909002098735962728218280573, 2.82502914032892148150823739661, 4.01944737301337215508374607197, 4.86392034667808960492565754334, 5.69515231175480476064251608929, 6.87503954975009162660418284946, 7.955045401605422795369110424804, 8.264355086141501641630741698175, 9.775436586645329878643515404986, 10.47193789352110275534540497930

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