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Properties

Label 2-882-9.4-c1-0-22
Degree $2$
Conductor $882$
Sign $0.957 - 0.287i$
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 + (0.349 − 1.69i)3-s + (−0.499 + 0.866i)4-s + (−0.794 + 1.37i)5-s + (1.64 − 0.545i)6-s − 0.999·8-s + (−2.75 − 1.18i)9-s − 1.58·10-s + (0.794 + 1.37i)11-s + (1.29 + 1.15i)12-s + (2.40 − 4.16i)13-s + (2.05 + 1.82i)15-s + (−0.5 − 0.866i)16-s + 5.39·17-s + (−0.349 − 2.97i)18-s + 7.09·19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.201 − 0.979i)3-s + (−0.249 + 0.433i)4-s + (−0.355 + 0.615i)5-s + (0.671 − 0.222i)6-s − 0.353·8-s + (−0.918 − 0.395i)9-s − 0.502·10-s + (0.239 + 0.414i)11-s + (0.373 + 0.332i)12-s + (0.667 − 1.15i)13-s + (0.530 + 0.472i)15-s + (−0.125 − 0.216i)16-s + 1.30·17-s + (−0.0824 − 0.702i)18-s + 1.62·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.957 - 0.287i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.957 - 0.287i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92105 + 0.281898i\)
\(L(\frac12)\) \(\approx\) \(1.92105 + 0.281898i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.349 + 1.69i)T \)
7 \( 1 \)
good5 \( 1 + (0.794 - 1.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.794 - 1.37i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.39T + 17T^{2} \)
19 \( 1 - 7.09T + 19T^{2} \)
23 \( 1 + (0.150 - 0.260i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.13 - 7.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.35 + 2.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.833 + 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.33 + 2.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.88T + 53T^{2} \)
59 \( 1 + (3.23 - 5.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.23 - 3.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.02 + 8.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + 16.0T + 73T^{2} \)
79 \( 1 + (4.19 + 7.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.18 - 2.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.21T + 89T^{2} \)
97 \( 1 + (-0.712 - 1.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.16442323942184788755429076022, −9.092529480619915411226892630965, −8.120620360088890172071114566432, −7.49967245537311892770264742991, −6.94536831169925314809437195642, −5.88475976273746809728681498779, −5.21015479924420109141562474898, −3.49564148819439181113393299090, −3.00900437444854330808116561329, −1.14794469468481374486503876194, 1.13928234191782547831801909414, 2.88341207738596044931861776223, 3.76583840480576380852483470540, 4.55834660338365512276120533092, 5.39983885414039468240922200254, 6.36352253335123679837282991747, 7.897393731816916295611781722122, 8.595723625151202996303373155072, 9.522399779967146923604699801362, 9.933613926538217087770779171100

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