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Properties

Label 2-882-63.4-c1-0-12
Degree $2$
Conductor $882$
Sign $-0.104 - 0.994i$
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 + (1.72 − 0.158i)3-s + (−0.499 − 0.866i)4-s − 3.44·5-s + (−0.724 + 1.57i)6-s + 0.999·8-s + (2.94 − 0.548i)9-s + (1.72 − 2.98i)10-s + 2·11-s + (−0.999 − 1.41i)12-s + (−2.44 + 4.24i)13-s + (−5.94 + 0.548i)15-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s + (−0.999 + 2.82i)18-s + (3.72 + 6.45i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.995 − 0.0917i)3-s + (−0.249 − 0.433i)4-s − 1.54·5-s + (−0.295 + 0.642i)6-s + 0.353·8-s + (0.983 − 0.182i)9-s + (0.545 − 0.944i)10-s + 0.603·11-s + (−0.288 − 0.408i)12-s + (−0.679 + 1.17i)13-s + (−1.53 + 0.141i)15-s + (−0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s + (−0.235 + 0.666i)18-s + (0.854 + 1.48i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.104 - 0.994i$
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: \((2,\ 882,\ (\ :1/2),\ -0.104 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.843951 + 0.937373i\)
\(L(\frac12)\) \(\approx\) \(0.843951 + 0.937373i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.158i)T \)
7 \( 1 \)
good5 \( 1 + 3.44T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (2.44 - 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.72 - 6.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + (1.44 + 2.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.89 - 6.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.89 - 8.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.44 - 2.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.89 - 8.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.550 + 0.953i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.72 + 9.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.55 - 2.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 + (-1.44 + 2.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.94 - 6.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.55 + 6.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.44 + 5.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.841334976116684518633963682034, −9.528688147750885737341256871993, −8.298880033479798770435406896173, −8.009432763789958892784222675753, −7.17128940635319226260268194013, −6.47791953933808369627268590669, −4.80708820189486857000569773561, −4.06061693216507061328071611463, −3.13426807534645963305105780383, −1.41204341373094505734167945390, 0.67666755754465585667376889540, 2.46861389423205526893037877598, 3.44374602909502597404019379723, 4.06839650972264514661832704195, 5.15097738331117285359535594193, 7.05295955465437418108956434790, 7.56766647938658013544021679933, 8.298196776053720589866170159070, 9.006923124194975506683036659064, 9.855903289858692660975419099855

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