Properties

Label 2-882-63.38-c1-0-28
Degree $2$
Conductor $882$
Sign $0.965 - 0.259i$
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 + (1.45 − 0.942i)3-s − 4-s + (−0.724 + 1.25i)5-s + (0.942 + 1.45i)6-s i·8-s + (1.22 − 2.73i)9-s + (−1.25 − 0.724i)10-s + (1.21 − 0.698i)11-s + (−1.45 + 0.942i)12-s + (3.03 − 1.75i)13-s + (0.129 + 2.50i)15-s + 16-s + (3.95 − 6.84i)17-s + (2.73 + 1.22i)18-s + (−3.61 + 2.08i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.839 − 0.544i)3-s − 0.5·4-s + (−0.324 + 0.561i)5-s + (0.384 + 0.593i)6-s − 0.353i·8-s + (0.408 − 0.912i)9-s + (−0.396 − 0.229i)10-s + (0.364 − 0.210i)11-s + (−0.419 + 0.272i)12-s + (0.841 − 0.485i)13-s + (0.0334 + 0.647i)15-s + 0.250·16-s + (0.958 − 1.66i)17-s + (0.645 + 0.288i)18-s + (−0.828 + 0.478i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.965 - 0.259i$
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: \((2,\ 882,\ (\ :1/2),\ 0.965 - 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99637 + 0.263717i\)
\(L(\frac12)\) \(\approx\) \(1.99637 + 0.263717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.45 + 0.942i)T \)
7 \( 1 \)
good5 \( 1 + (0.724 - 1.25i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.21 + 0.698i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.03 + 1.75i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.95 + 6.84i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.61 - 2.08i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.13 - 1.80i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.06 - 2.34i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.917iT - 31T^{2} \)
37 \( 1 + (-2.14 - 3.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.343 - 0.595i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.01 - 10.4i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.31T + 47T^{2} \)
53 \( 1 + (5.16 + 2.98i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.44T + 59T^{2} \)
61 \( 1 + 9.85iT - 61T^{2} \)
67 \( 1 + 2.97T + 67T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 + (9.79 + 5.65i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + (4.11 - 7.12i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.533 + 0.923i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.9 + 6.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.886069657121275618624027655571, −9.143461484824168156091651197383, −8.284473076098726555042359169633, −7.64470753116015217574581617846, −6.85972065800008290050157120053, −6.15001044500944277288465070635, −4.91574841531944548957438214559, −3.58143009438042716782768537999, −2.94012884041832907731370788205, −1.11543763667541119955679659662, 1.34884787164658893912004039400, 2.59035948113490285820147911557, 3.92627628332774425747705730825, 4.20776000284911057985239539314, 5.44885663806712581655204144773, 6.71776237928705291427862507358, 8.034641286093251996816606390665, 8.644084978450844676695511270250, 9.101420804533577646583676054812, 10.30875580931741481079806622612

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