Properties

Label 2-882-9.7-c1-0-23
Degree $2$
Conductor $882$
Sign $0.972 + 0.234i$
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 + (1.68 − 0.396i)3-s + (−0.499 − 0.866i)4-s + (2.18 + 3.78i)5-s + (0.5 − 1.65i)6-s − 0.999·8-s + (2.68 − 1.33i)9-s + 4.37·10-s + (0.686 − 1.18i)11-s + (−1.18 − 1.26i)12-s + (1 + 1.73i)13-s + (5.18 + 5.51i)15-s + (−0.5 + 0.866i)16-s − 1.37·17-s + (0.186 − 2.99i)18-s − 5·19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.973 − 0.228i)3-s + (−0.249 − 0.433i)4-s + (0.977 + 1.69i)5-s + (0.204 − 0.677i)6-s − 0.353·8-s + (0.895 − 0.445i)9-s + 1.38·10-s + (0.206 − 0.358i)11-s + (−0.342 − 0.364i)12-s + (0.277 + 0.480i)13-s + (1.33 + 1.42i)15-s + (−0.125 + 0.216i)16-s − 0.332·17-s + (0.0438 − 0.705i)18-s − 1.14·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.972 + 0.234i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.972 + 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.98076 - 0.354308i\)
\(L(\frac12)\) \(\approx\) \(2.98076 - 0.354308i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.396i)T \)
7 \( 1 \)
good5 \( 1 + (-2.18 - 3.78i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.686 + 1.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.37T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-0.813 - 1.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.37 + 7.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-2.31 - 4.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.05 + 7.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.74T + 53T^{2} \)
59 \( 1 + (5.05 + 8.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.55 + 2.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.05 - 1.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.11T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + (-2.55 + 4.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.74 + 15.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + (4.05 - 7.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.11994600910739909117356940616, −9.483033452925655667160339245092, −8.609331054237765520972309367685, −7.46317210003901296872235996682, −6.48516152979091867566085934991, −6.08847781245883529924651428174, −4.39399906929075744339790282848, −3.37529735395568641732892505039, −2.57263949076170193493565583571, −1.79625300960837753439898942385, 1.39937089212765376867117038282, 2.65610113886163924124538008959, 4.21758354051815543219830372636, 4.71058777897684283939515800266, 5.70098194972540099873915646448, 6.65584274935182745376749030947, 7.88592589589447604420591142984, 8.609953607084626181356670124542, 9.049446709516006186354418282196, 9.816528460635525447004251667780

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