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Properties

Label 2-882-441.142-c1-0-33
Degree $2$
Conductor $882$
Sign $0.999 - 0.000571i$
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  + (−0.365 + 0.930i)2-s + (−1.71 − 0.240i)3-s + (−0.733 − 0.680i)4-s + (3.67 − 1.76i)5-s + (0.850 − 1.50i)6-s + (2.48 − 0.907i)7-s + (0.900 − 0.433i)8-s + (2.88 + 0.826i)9-s + (0.304 + 4.06i)10-s + (1.23 + 1.54i)11-s + (1.09 + 1.34i)12-s + (−0.499 + 1.27i)13-s + (−0.0636 + 2.64i)14-s + (−6.72 + 2.14i)15-s + (0.0747 + 0.997i)16-s + (0.425 − 0.395i)17-s + ⋯
L(s)  = 1  + (−0.258 + 0.658i)2-s + (−0.990 − 0.139i)3-s + (−0.366 − 0.340i)4-s + (1.64 − 0.791i)5-s + (0.347 − 0.615i)6-s + (0.939 − 0.342i)7-s + (0.318 − 0.153i)8-s + (0.961 + 0.275i)9-s + (0.0963 + 1.28i)10-s + (0.372 + 0.466i)11-s + (0.315 + 0.387i)12-s + (−0.138 + 0.353i)13-s + (−0.0169 + 0.706i)14-s + (−1.73 + 0.555i)15-s + (0.0186 + 0.249i)16-s + (0.103 − 0.0958i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.000571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.000571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.999 - 0.000571i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.999 - 0.000571i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49149 + 0.000426429i\)
\(L(\frac12)\) \(\approx\) \(1.49149 + 0.000426429i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.365 - 0.930i)T \)
3 \( 1 + (1.71 + 0.240i)T \)
7 \( 1 + (-2.48 + 0.907i)T \)
good5 \( 1 + (-3.67 + 1.76i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-1.23 - 1.54i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.499 - 1.27i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-0.425 + 0.395i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-0.440 + 0.762i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.916 + 4.01i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.853 + 0.263i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (1.75 - 3.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.21 - 0.684i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (1.86 + 1.27i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (2.78 - 1.90i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (4.37 - 11.1i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-9.10 + 2.80i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (8.44 - 5.76i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (0.208 - 0.193i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-7.40 + 12.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.21 - 14.0i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-11.6 - 1.76i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (5.60 + 9.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.90 + 15.0i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (0.171 + 0.437i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-2.22 + 3.84i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.04935257095259841821399265596, −9.349513743753987587446678793702, −8.526637955278991402179625363877, −7.39782608230858045940994959993, −6.55542931561272041339277545158, −5.82498538332637960430719542152, −4.94714501468458910109146845235, −4.52124395118478746650314567432, −1.96613569950096862142691025172, −1.09970623286011468570807583587, 1.34020291974436997090097155512, 2.27382537196856383847325286188, 3.64007992457460758252361607783, 5.11582645746838932506778599241, 5.62320272506193485342465465179, 6.51129113874183137838756759263, 7.51239092730268187790507970711, 8.759191176366339753750159274649, 9.673666666230817453158249144974, 10.16315023620049402683363104381

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