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Properties

Label 2-882-63.38-c1-0-2
Degree $2$
Conductor $882$
Sign $0.577 + 0.816i$
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  + i·2-s + (−1.33 + 1.10i)3-s − 4-s + (−1.94 + 3.36i)5-s + (−1.10 − 1.33i)6-s − i·8-s + (0.577 − 2.94i)9-s + (−3.36 − 1.94i)10-s + (−3.41 + 1.97i)11-s + (1.33 − 1.10i)12-s + (−2.46 + 1.42i)13-s + (−1.10 − 6.64i)15-s + 16-s + (0.371 − 0.642i)17-s + (2.94 + 0.577i)18-s + (−1.54 + 0.892i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.772 + 0.635i)3-s − 0.5·4-s + (−0.870 + 1.50i)5-s + (−0.449 − 0.546i)6-s − 0.353i·8-s + (0.192 − 0.981i)9-s + (−1.06 − 0.615i)10-s + (−1.03 + 0.594i)11-s + (0.386 − 0.317i)12-s + (−0.684 + 0.395i)13-s + (−0.285 − 1.71i)15-s + 0.250·16-s + (0.0899 − 0.155i)17-s + (0.693 + 0.136i)18-s + (−0.354 + 0.204i)19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.139767 - 0.0723490i\)
\(L(\frac12)\) \(\approx\) \(0.139767 - 0.0723490i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (1.33 - 1.10i)T \)
7 \( 1 \)
good5 \( 1 + (1.94 - 3.36i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.41 - 1.97i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.46 - 1.42i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.371 + 0.642i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.54 - 0.892i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.41 - 3.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.50 + 1.44i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.51iT - 31T^{2} \)
37 \( 1 + (1.50 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.24 - 9.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.471 + 0.816i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.18T + 47T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.0211T + 59T^{2} \)
61 \( 1 + 2.46iT - 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 + (4.20 + 2.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.63T + 79T^{2} \)
83 \( 1 + (-4.02 + 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.63 + 8.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.2 + 9.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.87014924182272691665640256728, −10.03685405289797051767434405338, −9.388643239502616261873211738693, −7.976061308800300823083490113356, −7.28341148167837211380282726594, −6.71036160496562512290282925830, −5.68686606445607389335935946832, −4.72422883760869411832771758660, −3.84019304607112490385443230979, −2.72976787249408411583587856489, 0.099705584474472892653758285990, 1.08008766164946601273802717641, 2.60325010098881176887055236681, 4.07834904190475613169655844167, 5.12818142590117358356468112630, 5.39920424706014086215662387473, 6.97763139723354564782561320118, 7.962604726279402421671572981317, 8.454677204821366131595166315452, 9.404321657998615466353681923349

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