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Properties

Label 2-882-147.101-c1-0-2
Degree $2$
Conductor $882$
Sign $-0.752 - 0.658i$
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  + (0.930 + 0.365i)2-s + (0.733 + 0.680i)4-s + (−1.63 − 1.11i)5-s + (−2.45 − 0.975i)7-s + (0.433 + 0.900i)8-s + (−1.11 − 1.63i)10-s + (0.838 + 5.56i)11-s + (−2.27 + 1.81i)13-s + (−1.93 − 1.80i)14-s + (0.0747 + 0.997i)16-s + (−1.68 − 0.520i)17-s + (−6.29 + 3.63i)19-s + (−0.440 − 1.93i)20-s + (−1.25 + 5.48i)22-s + (1.94 + 6.31i)23-s + ⋯
L(s)  = 1  + (0.658 + 0.258i)2-s + (0.366 + 0.340i)4-s + (−0.732 − 0.499i)5-s + (−0.929 − 0.368i)7-s + (0.153 + 0.318i)8-s + (−0.352 − 0.517i)10-s + (0.252 + 1.67i)11-s + (−0.631 + 0.503i)13-s + (−0.516 − 0.482i)14-s + (0.0186 + 0.249i)16-s + (−0.409 − 0.126i)17-s + (−1.44 + 0.834i)19-s + (−0.0985 − 0.431i)20-s + (−0.266 + 1.16i)22-s + (0.406 + 1.31i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.752 - 0.658i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.335984 + 0.894336i\)
\(L(\frac12)\) \(\approx\) \(0.335984 + 0.894336i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.930 - 0.365i)T \)
3 \( 1 \)
7 \( 1 + (2.45 + 0.975i)T \)
good5 \( 1 + (1.63 + 1.11i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.838 - 5.56i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (2.27 - 1.81i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (1.68 + 0.520i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (6.29 - 3.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.94 - 6.31i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (-8.15 + 1.86i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-3.38 - 1.95i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.211 + 0.196i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (9.79 - 4.71i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (8.98 + 4.32i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (2.58 - 6.58i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (2.38 - 2.56i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-11.4 + 7.80i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (3.85 + 4.15i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (-3.06 + 5.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.16 + 0.951i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (8.67 - 3.40i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (5.42 + 9.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.38 + 5.50i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-11.0 - 1.67i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + 9.55iT - 97T^{2} \)
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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

−10.27166310918426926881917181972, −9.783390388001062497300279450850, −8.662486309526417077687289809748, −7.75330781538150202653197824814, −6.88006863120169442391695944464, −6.36894955073967844110843705159, −4.78934586986455342011065186866, −4.40867589771632626088428830670, −3.36137001750292960544240230782, −1.93200877996603433612633445533, 0.34624445990477572256781452789, 2.63384541222918543128195764628, 3.22587471739523909155051941025, 4.25558444465753962088710862280, 5.37355977422695489831901368544, 6.57245416651251040112712804668, 6.73251945594782507486948957981, 8.332138998700453171191188094864, 8.770148702759739728548299551255, 10.16850221318038856209520582600

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