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Properties

Label 2-882-441.130-c1-0-36
Degree $2$
Conductor $882$
Sign $-0.834 + 0.550i$
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  + (0.826 − 0.563i)2-s + (−1.43 − 0.969i)3-s + (0.365 − 0.930i)4-s + (0.633 + 2.77i)5-s + (−1.73 + 0.00729i)6-s + (−2.44 − 1.01i)7-s + (−0.222 − 0.974i)8-s + (1.11 + 2.78i)9-s + (2.08 + 1.93i)10-s + (−1.37 − 0.660i)11-s + (−1.42 + 0.981i)12-s + (3.74 − 2.55i)13-s + (−2.59 + 0.539i)14-s + (1.78 − 4.59i)15-s + (−0.733 − 0.680i)16-s + (−2.45 − 6.26i)17-s + ⋯
L(s)  = 1  + (0.584 − 0.398i)2-s + (−0.828 − 0.559i)3-s + (0.182 − 0.465i)4-s + (0.283 + 1.24i)5-s + (−0.707 + 0.00297i)6-s + (−0.923 − 0.383i)7-s + (−0.0786 − 0.344i)8-s + (0.373 + 0.927i)9-s + (0.660 + 0.612i)10-s + (−0.413 − 0.199i)11-s + (−0.411 + 0.283i)12-s + (1.03 − 0.707i)13-s + (−0.692 + 0.144i)14-s + (0.460 − 1.18i)15-s + (−0.183 − 0.170i)16-s + (−0.596 − 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.834 + 0.550i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.834 + 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.283245 - 0.944548i\)
\(L(\frac12)\) \(\approx\) \(0.283245 - 0.944548i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.826 + 0.563i)T \)
3 \( 1 + (1.43 + 0.969i)T \)
7 \( 1 + (2.44 + 1.01i)T \)
good5 \( 1 + (-0.633 - 2.77i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (1.37 + 0.660i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-3.74 + 2.55i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (2.45 + 6.26i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (0.281 + 0.486i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.21 + 6.53i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (6.65 + 1.00i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (0.291 + 0.505i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.97 + 0.900i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (-8.77 - 2.70i)T + (33.8 + 23.0i)T^{2} \)
43 \( 1 + (-4.22 + 1.30i)T + (35.5 - 24.2i)T^{2} \)
47 \( 1 + (-5.92 + 4.03i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (9.03 - 1.36i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (7.44 - 2.29i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-2.83 - 7.22i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-0.591 - 1.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.26 + 4.08i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.766 + 10.2i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (-3.63 + 6.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.90 - 1.29i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (-1.82 - 1.24i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (-0.0687 - 0.119i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.32347996030388791565888428804, −9.212831603888543902964442696604, −7.70557804187601835551727166711, −6.95083036517381805282559519680, −6.22453958729983146117395096752, −5.68654476117036708133173830470, −4.34865456120400342390767033033, −3.14572420314088233576742683901, −2.30977028890001319307486807529, −0.41817784596640579592470310258, 1.69639425909202841447595759900, 3.68222678193995909170239882802, 4.23207430741754414235813742194, 5.44447073489574393968555913953, 5.88033975221095045142275672304, 6.63642497790061205725687254804, 7.963105504209760467948720038183, 9.051148637722798621803773127578, 9.404524517749426886224141550192, 10.56442390698534048821439453250

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