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Properties

Label 2-882-63.38-c1-0-37
Degree $2$
Conductor $882$
Sign $-0.984 + 0.173i$
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 + (0.541 − 1.64i)3-s − 4-s + (0.895 − 1.55i)5-s + (−1.64 − 0.541i)6-s + i·8-s + (−2.41 − 1.78i)9-s + (−1.55 − 0.895i)10-s + (2.07 − 1.20i)11-s + (−0.541 + 1.64i)12-s + (4.23 − 2.44i)13-s + (−2.06 − 2.31i)15-s + 16-s + (1.83 − 3.17i)17-s + (−1.78 + 2.41i)18-s + (−2.61 + 1.50i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.312 − 0.949i)3-s − 0.5·4-s + (0.400 − 0.693i)5-s + (−0.671 − 0.220i)6-s + 0.353i·8-s + (−0.804 − 0.593i)9-s + (−0.490 − 0.283i)10-s + (0.627 − 0.362i)11-s + (−0.156 + 0.474i)12-s + (1.17 − 0.678i)13-s + (−0.533 − 0.596i)15-s + 0.250·16-s + (0.444 − 0.769i)17-s + (−0.419 + 0.569i)18-s + (−0.599 + 0.346i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.984 + 0.173i$
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: \((2,\ 882,\ (\ :1/2),\ -0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147639 - 1.69387i\)
\(L(\frac12)\) \(\approx\) \(0.147639 - 1.69387i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.541 + 1.64i)T \)
7 \( 1 \)
good5 \( 1 + (-0.895 + 1.55i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.07 + 1.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.23 + 2.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.83 + 3.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.61 - 1.50i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.26 + 1.88i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.68 + 3.28i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.64iT - 31T^{2} \)
37 \( 1 + (4.68 + 8.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.04 - 6.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.48 - 6.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.13T + 47T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 - 11.3iT - 61T^{2} \)
67 \( 1 - 0.570T + 67T^{2} \)
71 \( 1 - 5.96iT - 71T^{2} \)
73 \( 1 + (10.7 + 6.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.03T + 79T^{2} \)
83 \( 1 + (-7.00 + 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.87 + 3.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.77 + 2.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.612589020684191609960864974179, −8.825973384013199703174306464001, −8.335677764966139408915075964295, −7.30552644619579218151204050803, −6.09436069053929841023641237425, −5.50673446711159546422948169541, −4.04789196690621937145031027740, −3.07410442205656759144858619672, −1.78732735211731380114650224022, −0.833071384143681845729544972047, 2.01892135632647361520354248230, 3.59153108674749287084037424509, 4.12485028019694314396035379118, 5.40502306967184332408135348206, 6.21895566543833055615409951292, 6.97395284521681616885299717384, 8.180377500587643478830642279641, 8.847699493071481148606177189377, 9.609662076907435090335394838655, 10.39576946610715933604781651706

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