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Properties

Label 2-882-63.16-c1-0-1
Degree $2$
Conductor $882$
Sign $0.382 - 0.923i$
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  + (−0.5 − 0.866i)2-s + (−1.73 − 0.0789i)3-s + (−0.499 + 0.866i)4-s + 0.460·5-s + (0.796 + 1.53i)6-s + 0.999·8-s + (2.98 + 0.273i)9-s + (−0.230 − 0.398i)10-s − 3.64·11-s + (0.933 − 1.45i)12-s + (−0.730 − 1.26i)13-s + (−0.796 − 0.0363i)15-s + (−0.5 − 0.866i)16-s + (1.86 + 3.23i)17-s + (−1.25 − 2.72i)18-s + (2.02 − 3.51i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.998 − 0.0455i)3-s + (−0.249 + 0.433i)4-s + 0.205·5-s + (0.325 + 0.627i)6-s + 0.353·8-s + (0.995 + 0.0910i)9-s + (−0.0728 − 0.126i)10-s − 1.09·11-s + (0.269 − 0.421i)12-s + (−0.202 − 0.350i)13-s + (−0.205 − 0.00938i)15-s + (−0.125 − 0.216i)16-s + (0.452 + 0.784i)17-s + (−0.296 − 0.642i)18-s + (0.465 − 0.805i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.406538 + 0.271719i\)
\(L(\frac12)\) \(\approx\) \(0.406538 + 0.271719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (1.73 + 0.0789i)T \)
7 \( 1 \)
good5 \( 1 - 0.460T + 5T^{2} \)
11 \( 1 + 3.64T + 11T^{2} \)
13 \( 1 + (0.730 + 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.86 - 3.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.02 + 3.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.13T + 23T^{2} \)
29 \( 1 + (4.48 - 7.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.257 - 0.445i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.55 - 7.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.472 - 0.819i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.66 + 8.07i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.16 - 2.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.21 - 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.44 - 11.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.04 - 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.16 + 2.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.67T + 71T^{2} \)
73 \( 1 + (-6.62 - 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.50 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.32 - 5.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.36 + 2.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.59 + 9.68i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.48412767438222216271979196197, −9.748701497582731722866269717208, −8.751023505289475518560673581000, −7.67946873510225365473848893888, −7.01991189373291844939374825637, −5.69287179745178619441316319072, −5.16452201454441146697618866108, −3.95041317739410855247676501935, −2.66120226395687276769610429365, −1.27060933776965296656775755259, 0.32557833121782557241392061975, 2.03666616262689000426083828359, 3.83589118059915287949909657159, 5.07459244940826956850564310149, 5.58873781395841105579615230220, 6.46123453408454144058833791969, 7.50699570318097578126151368427, 7.942296863302991506901589061739, 9.457826073428338545642448874663, 9.811130035695542935967077025660

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