Properties

Label 2-882-441.16-c1-0-47
Degree $2$
Conductor $882$
Sign $0.781 + 0.623i$
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  + (0.988 + 0.149i)2-s + (1.71 + 0.242i)3-s + (0.955 + 0.294i)4-s + (−2.59 − 1.24i)5-s + (1.65 + 0.495i)6-s + (−1.92 − 1.81i)7-s + (0.900 + 0.433i)8-s + (2.88 + 0.832i)9-s + (−2.37 − 1.61i)10-s + (2.57 − 3.22i)11-s + (1.56 + 0.737i)12-s + (1.21 + 0.183i)13-s + (−1.63 − 2.08i)14-s + (−4.14 − 2.76i)15-s + (0.826 + 0.563i)16-s + (3.84 − 1.18i)17-s + ⋯
L(s)  = 1  + (0.699 + 0.105i)2-s + (0.990 + 0.140i)3-s + (0.477 + 0.147i)4-s + (−1.15 − 0.557i)5-s + (0.677 + 0.202i)6-s + (−0.727 − 0.685i)7-s + (0.318 + 0.153i)8-s + (0.960 + 0.277i)9-s + (−0.751 − 0.512i)10-s + (0.775 − 0.971i)11-s + (0.452 + 0.212i)12-s + (0.337 + 0.0508i)13-s + (−0.436 − 0.556i)14-s + (−1.06 − 0.714i)15-s + (0.206 + 0.140i)16-s + (0.933 − 0.287i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.781 + 0.623i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.781 + 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60539 - 0.912129i\)
\(L(\frac12)\) \(\approx\) \(2.60539 - 0.912129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.988 - 0.149i)T \)
3 \( 1 + (-1.71 - 0.242i)T \)
7 \( 1 + (1.92 + 1.81i)T \)
good5 \( 1 + (2.59 + 1.24i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-2.57 + 3.22i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-1.21 - 0.183i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (-3.84 + 1.18i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-2.59 + 4.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.31 - 5.74i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (5.92 + 5.50i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (0.462 - 0.801i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.81 - 3.53i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (0.697 + 9.30i)T + (-40.5 + 6.11i)T^{2} \)
43 \( 1 + (0.347 - 4.64i)T + (-42.5 - 6.40i)T^{2} \)
47 \( 1 + (5.91 + 0.892i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (8.71 - 8.08i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (-0.583 + 7.78i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-9.63 + 2.97i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (6.19 - 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.855 - 3.74i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (4.37 - 11.1i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (-3.84 - 6.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.14 - 0.926i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (5.93 - 0.894i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (7.14 - 12.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.836781568798211998916599137144, −9.189215978689957340252780594031, −8.200432563712813198951973918762, −7.53923623265878107217209449513, −6.80156195323731134289378686306, −5.53061714134368045214193559719, −4.29844847657381547315209251466, −3.66440620933679553734139179209, −3.06829459659800028306131185106, −1.06116837206489405976839570182, 1.76998037355729106576753214514, 3.17789678177516901541193065071, 3.56219939532558544804649644780, 4.56127422031093879679296528089, 5.99715003325390095682513551212, 6.91577897975170983211771343262, 7.57254655540612178910049212312, 8.416751857082496862968866207860, 9.456990055799978057302444563796, 10.12259845355014961649845622167

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