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Properties

Label 2-882-63.20-c1-0-21
Degree $2$
Conductor $882$
Sign $-0.118 - 0.992i$
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  + (0.866 + 0.5i)2-s + (1.28 + 1.16i)3-s + (0.499 + 0.866i)4-s + (0.450 + 0.779i)5-s + (0.533 + 1.64i)6-s + 0.999i·8-s + (0.308 + 2.98i)9-s + 0.900i·10-s + (2.70 + 1.56i)11-s + (−0.361 + 1.69i)12-s + (1.99 − 1.14i)13-s + (−0.325 + 1.52i)15-s + (−0.5 + 0.866i)16-s − 5.15·17-s + (−1.22 + 2.73i)18-s − 2.74i·19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.742 + 0.669i)3-s + (0.249 + 0.433i)4-s + (0.201 + 0.348i)5-s + (0.217 + 0.672i)6-s + 0.353i·8-s + (0.102 + 0.994i)9-s + 0.284i·10-s + (0.816 + 0.471i)11-s + (−0.104 + 0.488i)12-s + (0.552 − 0.318i)13-s + (−0.0840 + 0.393i)15-s + (−0.125 + 0.216i)16-s − 1.24·17-s + (−0.288 + 0.645i)18-s − 0.630i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.118 - 0.992i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.118 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98238 + 2.23262i\)
\(L(\frac12)\) \(\approx\) \(1.98238 + 2.23262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-1.28 - 1.16i)T \)
7 \( 1 \)
good5 \( 1 + (-0.450 - 0.779i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.70 - 1.56i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.99 + 1.14i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.15T + 17T^{2} \)
19 \( 1 + 2.74iT - 19T^{2} \)
23 \( 1 + (-1.48 + 0.857i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.85 + 1.07i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.66 - 5.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.47T + 37T^{2} \)
41 \( 1 + (1.22 + 2.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.273 - 0.473i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.93 + 6.80i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 13.9iT - 53T^{2} \)
59 \( 1 + (-3.99 - 6.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.28 + 3.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.83 + 3.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 + (-3.27 + 5.67i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.184 - 0.319i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (8.86 + 5.12i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.54979448111170542079888210155, −9.166343697197955903279780914809, −8.946594268791188841466652933506, −7.73672884010261602497210765497, −6.91078767809104605324773131284, −6.03727930266735388680528326651, −4.82365412821909214883670031306, −4.13182025286398108780207731661, −3.12572417579885815315808578264, −2.07286702692552144011863249123, 1.21528612686041622708283754331, 2.23198702496750843319658778816, 3.50714590215879572041658242938, 4.22095205746676741332414947567, 5.61579400515637402175584616328, 6.43093986785098084165709478868, 7.21984145798884705521272357445, 8.350365018176095638400880270782, 9.109637500216336164515126672084, 9.666515070423246977106748866963

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