Properties

Label 2-882-3.2-c4-0-36
Degree $2$
Conductor $882$
Sign $0.816 + 0.577i$
Analytic cond. $91.1723$
Root an. cond. $9.54841$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s − 8.00·4-s + 27.6i·5-s − 22.6i·8-s − 78.1·10-s + 68.7i·11-s − 234.·13-s + 64.0·16-s + 232. i·17-s − 358.·19-s − 220. i·20-s − 194.·22-s − 673. i·23-s − 137.·25-s − 661. i·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s + 1.10i·5-s − 0.353i·8-s − 0.781·10-s + 0.568i·11-s − 1.38·13-s + 0.250·16-s + 0.803i·17-s − 0.992·19-s − 0.552i·20-s − 0.401·22-s − 1.27i·23-s − 0.220·25-s − 0.979i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(91.1723\)
Root analytic conductor: \(9.54841\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :2),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.3672691074\)
\(L(\frac12)\) \(\approx\) \(0.3672691074\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 27.6iT - 625T^{2} \)
11 \( 1 - 68.7iT - 1.46e4T^{2} \)
13 \( 1 + 234.T + 2.85e4T^{2} \)
17 \( 1 - 232. iT - 8.35e4T^{2} \)
19 \( 1 + 358.T + 1.30e5T^{2} \)
23 \( 1 + 673. iT - 2.79e5T^{2} \)
29 \( 1 - 791. iT - 7.07e5T^{2} \)
31 \( 1 - 705.T + 9.23e5T^{2} \)
37 \( 1 + 1.37e3T + 1.87e6T^{2} \)
41 \( 1 + 56.5iT - 2.82e6T^{2} \)
43 \( 1 - 263.T + 3.41e6T^{2} \)
47 \( 1 + 2.28e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.32e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.05e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.58e3T + 1.38e7T^{2} \)
67 \( 1 + 5.83e3T + 2.01e7T^{2} \)
71 \( 1 + 8.12e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.19e3T + 2.83e7T^{2} \)
79 \( 1 - 1.05e4T + 3.89e7T^{2} \)
83 \( 1 - 7.94e3iT - 4.74e7T^{2} \)
89 \( 1 - 6.28e3iT - 6.27e7T^{2} \)
97 \( 1 + 852.T + 8.85e7T^{2} \)
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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

−9.454065314572263521640392136212, −8.506509661141081768002017018411, −7.61645816040894398598208032496, −6.81800206147210359448496505008, −6.35973686201132004560997047331, −5.08821400470585506257544885431, −4.26713487828885834689359960990, −3.02073605607790130468793861299, −2.00297621543008021574401088452, −0.098490862327920629275446187096, 0.859486867975118897449374699638, 2.04765591294413404722152372117, 3.12741784811143908481558938539, 4.40011448686517570763181365757, 4.98623878650201094408866861002, 5.94280519589832421077683676076, 7.28132669815846325253208264851, 8.162002447631974086077996941985, 8.999186076066552089711924263573, 9.601658141535603162139979786051

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