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Properties

Label 2-882-7.4-c3-0-47
Degree $2$
Conductor $882$
Sign $-0.386 + 0.922i$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (−6 − 10.3i)5-s − 7.99·8-s + (12 − 20.7i)10-s + (24 − 41.5i)11-s + 56·13-s + (−8 − 13.8i)16-s + (−57 + 98.7i)17-s + (−1 − 1.73i)19-s + 48·20-s + 96·22-s + (−60 − 103. i)23-s + (−9.5 + 16.4i)25-s + (56 + 96.9i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.536 − 0.929i)5-s − 0.353·8-s + (0.379 − 0.657i)10-s + (0.657 − 1.13i)11-s + 1.19·13-s + (−0.125 − 0.216i)16-s + (−0.813 + 1.40i)17-s + (−0.0120 − 0.0209i)19-s + 0.536·20-s + 0.930·22-s + (−0.543 − 0.942i)23-s + (−0.0759 + 0.131i)25-s + (0.422 + 0.731i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9674572976\)
\(L(\frac12)\) \(\approx\) \(0.9674572976\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (6 + 10.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-24 + 41.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 56T + 2.19e3T^{2} \)
17 \( 1 + (57 - 98.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (60 + 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 54T + 2.43e4T^{2} \)
31 \( 1 + (118 - 204. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (73 + 126. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 126T + 6.89e4T^{2} \)
43 \( 1 + 376T + 7.95e4T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-87 + 150. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-69 + 119. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (190 + 329. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-242 + 419. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 576T + 3.57e5T^{2} \)
73 \( 1 + (-575 + 995. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (388 + 672. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 378T + 5.71e5T^{2} \)
89 \( 1 + (195 + 337. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.33e3T + 9.12e5T^{2} \)
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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

−8.922553411953916553859947489529, −8.606168293349679099328712125878, −8.062528080053267827281816832176, −6.61565577414794532100974835964, −6.14910841885089154079067712168, −5.05181550809757590497152944516, −4.06879366356212266112056526597, −3.44784384529873341657743074169, −1.54290879248312543222384624022, −0.22635467242426747302159979827, 1.44557325087370246360990278162, 2.63490221577512492950750090011, 3.66008697534077786369446962037, 4.37761479730857268629575623636, 5.56275776426569389861747546061, 6.73063353873411338409650450755, 7.21201218608357151135395158566, 8.414768520957178967010800674330, 9.430103555112888622989388064035, 10.04424337270821070038079942943

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