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Properties

Label 2-882-63.4-c1-0-27
Degree $2$
Conductor $882$
Sign $-0.608 + 0.793i$
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 + (−0.796 + 1.53i)3-s + (−0.499 − 0.866i)4-s − 0.593·5-s + (0.933 + 1.45i)6-s − 0.999·8-s + (−1.73 − 2.45i)9-s + (−0.296 + 0.514i)10-s − 0.593·11-s + (1.73 − 0.0789i)12-s + (1.25 − 2.17i)13-s + (0.472 − 0.912i)15-s + (−0.5 + 0.866i)16-s + (−1.46 + 2.52i)17-s + (−2.98 + 0.273i)18-s + (−2.69 − 4.66i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.460 + 0.887i)3-s + (−0.249 − 0.433i)4-s − 0.265·5-s + (0.381 + 0.595i)6-s − 0.353·8-s + (−0.576 − 0.816i)9-s + (−0.0938 + 0.162i)10-s − 0.178·11-s + (0.499 − 0.0227i)12-s + (0.348 − 0.603i)13-s + (0.122 − 0.235i)15-s + (−0.125 + 0.216i)16-s + (−0.354 + 0.613i)17-s + (−0.704 + 0.0643i)18-s + (−0.617 − 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.608 + 0.793i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.352078 - 0.714185i\)
\(L(\frac12)\) \(\approx\) \(0.352078 - 0.714185i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.796 - 1.53i)T \)
7 \( 1 \)
good5 \( 1 + 0.593T + 5T^{2} \)
11 \( 1 + 0.593T + 11T^{2} \)
13 \( 1 + (-1.25 + 2.17i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.46 - 2.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.69 + 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.46T + 23T^{2} \)
29 \( 1 + (3.09 + 5.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.93 + 6.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.136 + 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.58 + 9.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.08 + 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.02 + 6.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.32 - 7.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.32 - 5.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.956 - 1.65i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (3.95 - 6.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.62 + 8.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.85 + 6.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.21 - 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.86 + 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.14570812906514688203649201206, −9.128727702070625757448230636540, −8.490700742993706447064813578370, −7.17504324354927848316463890733, −6.00426852614445019466856811867, −5.33008030029587114905917861928, −4.26082429886164228258604970531, −3.62640808473415162299507896563, −2.35122072651991706054154157876, −0.35676202302856068109599730474, 1.60982015257796281115018729157, 3.09536346883604719817534817938, 4.37771959156437190318007914439, 5.35354772291003960569278005938, 6.20104605944529926894323329318, 6.98389898657521094439239212326, 7.66855486851094427028996958333, 8.511365369229489965549495203750, 9.352605337987990635341621962292, 10.71903823938550972077457143256

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