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Properties

Label 2-882-441.142-c1-0-10
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  + (−0.365 + 0.930i)2-s + (−1.66 + 0.475i)3-s + (−0.733 − 0.680i)4-s + (−2.16 + 1.04i)5-s + (0.165 − 1.72i)6-s + (0.529 + 2.59i)7-s + (0.900 − 0.433i)8-s + (2.54 − 1.58i)9-s + (−0.179 − 2.39i)10-s + (3.71 + 4.65i)11-s + (1.54 + 0.783i)12-s + (−1.01 + 2.57i)13-s + (−2.60 − 0.454i)14-s + (3.11 − 2.76i)15-s + (0.0747 + 0.997i)16-s + (3.21 − 2.98i)17-s + ⋯
L(s)  = 1  + (−0.258 + 0.658i)2-s + (−0.961 + 0.274i)3-s + (−0.366 − 0.340i)4-s + (−0.969 + 0.466i)5-s + (0.0675 − 0.703i)6-s + (0.200 + 0.979i)7-s + (0.318 − 0.153i)8-s + (0.849 − 0.528i)9-s + (−0.0568 − 0.758i)10-s + (1.11 + 1.40i)11-s + (0.445 + 0.226i)12-s + (−0.280 + 0.714i)13-s + (−0.696 − 0.121i)14-s + (0.803 − 0.715i)15-s + (0.0186 + 0.249i)16-s + (0.779 − 0.723i)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: $\chi_{882} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.858 + 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.155532 - 0.562851i\)
\(L(\frac12)\) \(\approx\) \(0.155532 - 0.562851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 - 0.475i)T \)
7 \( 1 + (-0.529 - 2.59i)T \)
good5 \( 1 + (2.16 - 1.04i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-3.71 - 4.65i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.01 - 2.57i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-3.21 + 2.98i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (3.75 - 6.50i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0371 - 0.162i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-8.62 - 2.66i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-1.70 + 2.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.81 + 2.41i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (6.62 + 4.51i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (3.86 - 2.63i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-2.11 + 5.37i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-2.33 + 0.719i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (0.899 - 0.613i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (3.33 - 3.09i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-7.30 + 12.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.33 - 10.2i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (0.926 + 0.139i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-0.765 - 1.32i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.56 + 3.99i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-0.879 - 2.24i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (2.57 - 4.45i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.45633105470695271146143659118, −9.819116613086204110181638681219, −8.973465620720390169242507923707, −7.967312685115875847062217018737, −6.98332610173241606425652448798, −6.55965314825170093700847468535, −5.42176770706834855697037420873, −4.55832381595491090073185882783, −3.71664393753474593068606953921, −1.72730831298881597902761385675, 0.41769906345791720366717100618, 1.21515355150064654164314544551, 3.26550848138183147351406348437, 4.21080474281777104163106724324, 4.94440992715276362927631348782, 6.27540682616811882762811798612, 7.07618301939203043452411293791, 8.198841793347445478370267525219, 8.555661781944000356551472122745, 10.00442789924651270764256752343

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