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Properties

Label 2-882-441.122-c1-0-52
Degree $2$
Conductor $882$
Sign $-0.980 + 0.197i$
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  + (0.680 − 0.733i)2-s + (1.33 − 1.09i)3-s + (−0.0747 − 0.997i)4-s + (−0.764 + 0.958i)5-s + (0.105 − 1.72i)6-s + (−2.62 − 0.342i)7-s + (−0.781 − 0.623i)8-s + (0.587 − 2.94i)9-s + (0.182 + 1.21i)10-s + (−5.88 − 1.34i)11-s + (−1.19 − 1.25i)12-s + (2.69 − 2.90i)13-s + (−2.03 + 1.69i)14-s + (0.0288 + 2.12i)15-s + (−0.988 + 0.149i)16-s + (0.143 − 1.91i)17-s + ⋯
L(s)  = 1  + (0.480 − 0.518i)2-s + (0.773 − 0.634i)3-s + (−0.0373 − 0.498i)4-s + (−0.341 + 0.428i)5-s + (0.0432 − 0.705i)6-s + (−0.991 − 0.129i)7-s + (−0.276 − 0.220i)8-s + (0.195 − 0.980i)9-s + (0.0577 + 0.383i)10-s + (−1.77 − 0.404i)11-s + (−0.345 − 0.361i)12-s + (0.746 − 0.805i)13-s + (−0.543 + 0.451i)14-s + (0.00744 + 0.548i)15-s + (−0.247 + 0.0372i)16-s + (0.0348 − 0.465i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.980 + 0.197i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.980 + 0.197i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147018 - 1.47511i\)
\(L(\frac12)\) \(\approx\) \(0.147018 - 1.47511i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.680 + 0.733i)T \)
3 \( 1 + (-1.33 + 1.09i)T \)
7 \( 1 + (2.62 + 0.342i)T \)
good5 \( 1 + (0.764 - 0.958i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (5.88 + 1.34i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-2.69 + 2.90i)T + (-0.971 - 12.9i)T^{2} \)
17 \( 1 + (-0.143 + 1.91i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (4.52 - 2.61i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.23 + 6.72i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (-1.11 + 1.63i)T + (-10.5 - 26.9i)T^{2} \)
31 \( 1 + (0.950 - 0.549i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.411 + 0.280i)T + (13.5 + 34.4i)T^{2} \)
41 \( 1 + (-2.14 - 5.45i)T + (-30.0 + 27.8i)T^{2} \)
43 \( 1 + (-3.81 + 9.72i)T + (-31.5 - 29.2i)T^{2} \)
47 \( 1 + (1.39 + 1.29i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (4.99 + 7.32i)T + (-19.3 + 49.3i)T^{2} \)
59 \( 1 + (-0.444 + 1.13i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (-13.0 - 0.981i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (-5.28 - 9.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.91 - 8.12i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (-3.13 + 10.1i)T + (-60.3 - 41.1i)T^{2} \)
79 \( 1 + (7.76 - 13.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.07 + 4.71i)T + (6.20 - 82.7i)T^{2} \)
89 \( 1 + (-2.91 + 2.70i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + (-2.71 + 1.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.00958471898035978222280986433, −8.765890271980243977915011009527, −8.125817901834453648581598264837, −7.14360743659827353124870140729, −6.32690173790798358128951337905, −5.37034630432383468878992070946, −3.89628444960759662779947356545, −3.05947809103977996153992402423, −2.45417904690998828700404146493, −0.51772946526664962002679408957, 2.36175984180787716016395322085, 3.35022749519135142505581010832, 4.29924639201008672461601324318, 5.08746290280380915199146256450, 6.14103046236700764303766624530, 7.23099811089599111531463999073, 8.033333882192089270067269879078, 8.797202180555080361560647859769, 9.487493318829935778466663907876, 10.46792417766868137394770895449

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