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Properties

Label 2-882-63.59-c1-0-34
Degree $2$
Conductor $882$
Sign $-0.734 + 0.679i$
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 + (1.15 − 1.29i)3-s + (0.499 + 0.866i)4-s − 1.79·5-s + (−1.64 + 0.541i)6-s − 0.999i·8-s + (−0.334 − 2.98i)9-s + (1.55 + 0.895i)10-s − 2.40i·11-s + (1.69 + 0.354i)12-s + (4.23 + 2.44i)13-s + (−2.06 + 2.31i)15-s + (−0.5 + 0.866i)16-s + (1.83 − 3.17i)17-s + (−1.20 + 2.74i)18-s + (2.61 − 1.50i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.666 − 0.745i)3-s + (0.249 + 0.433i)4-s − 0.800·5-s + (−0.671 + 0.220i)6-s − 0.353i·8-s + (−0.111 − 0.993i)9-s + (0.490 + 0.283i)10-s − 0.724i·11-s + (0.489 + 0.102i)12-s + (1.17 + 0.678i)13-s + (−0.533 + 0.596i)15-s + (−0.125 + 0.216i)16-s + (0.444 − 0.769i)17-s + (−0.283 + 0.647i)18-s + (0.599 − 0.346i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.734 + 0.679i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.734 + 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.395894 - 1.01085i\)
\(L(\frac12)\) \(\approx\) \(0.395894 - 1.01085i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 + 1.29i)T \)
7 \( 1 \)
good5 \( 1 + 1.79T + 5T^{2} \)
11 \( 1 + 2.40iT - 11T^{2} \)
13 \( 1 + (-4.23 - 2.44i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.83 + 3.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.61 + 1.50i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.76iT - 23T^{2} \)
29 \( 1 + (5.68 - 3.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.02 - 2.32i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.68 + 8.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.04 + 6.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.48 + 6.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.56 - 4.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.29 + 12.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.81 - 5.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.285 + 0.493i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.96iT - 71T^{2} \)
73 \( 1 + (-10.7 - 6.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.51 - 2.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.00 - 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.87 + 3.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.77 - 2.75i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.493692321810407646540362353150, −8.895019618765333115826471485756, −8.245223633834605357977590468394, −7.40070794408266572321626795508, −6.78261367314303552911550688012, −5.57976680380189603309397891883, −3.85644810721613492853977188584, −3.31616644796556663106954728135, −1.95305801681703617209984083284, −0.60802089326901471905585985187, 1.64686520201924475135242736994, 3.28556801858819535673069160062, 3.98282381070381630739278900174, 5.19618598448459519957212777658, 6.14786089507064445208349168262, 7.58456771488293458769922476881, 7.889539802192357899240192291742, 8.695445474528670567692299965475, 9.643789402895483540828207953263, 10.17773032822835838625620581134

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