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Properties

Label 2-882-441.110-c1-0-55
Degree $2$
Conductor $882$
Sign $-0.858 - 0.513i$
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  + (0.997 − 0.0747i)2-s + (−0.688 − 1.58i)3-s + (0.988 − 0.149i)4-s + (−0.0503 − 0.220i)5-s + (−0.804 − 1.53i)6-s + (−2.35 − 1.20i)7-s + (0.974 − 0.222i)8-s + (−2.05 + 2.18i)9-s + (−0.0666 − 0.216i)10-s + (−1.68 + 3.49i)11-s + (−0.917 − 1.46i)12-s + (−6.24 + 0.468i)13-s + (−2.43 − 1.02i)14-s + (−0.315 + 0.231i)15-s + (0.955 − 0.294i)16-s + (−3.98 − 0.601i)17-s + ⋯
L(s)  = 1  + (0.705 − 0.0528i)2-s + (−0.397 − 0.917i)3-s + (0.494 − 0.0745i)4-s + (−0.0225 − 0.0986i)5-s + (−0.328 − 0.626i)6-s + (−0.890 − 0.455i)7-s + (0.344 − 0.0786i)8-s + (−0.684 + 0.729i)9-s + (−0.0210 − 0.0683i)10-s + (−0.507 + 1.05i)11-s + (−0.264 − 0.424i)12-s + (−1.73 + 0.129i)13-s + (−0.651 − 0.274i)14-s + (−0.0815 + 0.0598i)15-s + (0.238 − 0.0736i)16-s + (−0.967 − 0.145i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.858 - 0.513i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.858 - 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0722018 + 0.261428i\)
\(L(\frac12)\) \(\approx\) \(0.0722018 + 0.261428i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.997 + 0.0747i)T \)
3 \( 1 + (0.688 + 1.58i)T \)
7 \( 1 + (2.35 + 1.20i)T \)
good5 \( 1 + (0.0503 + 0.220i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (1.68 - 3.49i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (6.24 - 0.468i)T + (12.8 - 1.93i)T^{2} \)
17 \( 1 + (3.98 + 0.601i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (1.66 + 0.963i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.659 - 0.526i)T + (5.11 - 22.4i)T^{2} \)
29 \( 1 + (-5.53 + 2.17i)T + (21.2 - 19.7i)T^{2} \)
31 \( 1 + (1.65 + 0.955i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.25 - 3.19i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (3.04 - 2.82i)T + (3.06 - 40.8i)T^{2} \)
43 \( 1 + (7.87 + 7.30i)T + (3.21 + 42.8i)T^{2} \)
47 \( 1 + (0.698 + 9.31i)T + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (7.70 + 3.02i)T + (38.8 + 36.0i)T^{2} \)
59 \( 1 + (0.170 + 0.157i)T + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (1.14 - 7.59i)T + (-58.2 - 17.9i)T^{2} \)
67 \( 1 + (4.98 - 8.62i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.51 + 5.19i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (4.49 - 6.58i)T + (-26.6 - 67.9i)T^{2} \)
79 \( 1 + (7.01 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.956 + 12.7i)T + (-82.0 - 12.3i)T^{2} \)
89 \( 1 + (-0.0644 + 0.860i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (9.00 + 5.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.00492661717903446853672199989, −8.697989420685654342397887057055, −7.49490837277649228085086429090, −6.96562568731348506298188085151, −6.36971926605337074718499200606, −5.05784794066946257678582705132, −4.52000055122237469283105381855, −2.87757037301096123005552206691, −2.07011034404049334068477323807, −0.096288716663886413144611433062, 2.65725138781032987395389471326, 3.28840609393825794072423474412, 4.56765408042779189650168139552, 5.23387287117570101375449819257, 6.17672675314414363467183621102, 6.84043410660953659877580083115, 8.169803660848161778056884519546, 9.125427064797636072126438716920, 9.899651451683205225071761911624, 10.72261418874610736918945621717

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