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Properties

Label 2-882-63.16-c1-0-0
Degree $2$
Conductor $882$
Sign $-0.971 - 0.236i$
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 − 1.37·5-s + (−0.5 − 1.65i)6-s − 0.999·8-s + (2.68 + 1.33i)9-s + (−0.686 − 1.18i)10-s + 4.37·11-s + (1.18 − 1.26i)12-s + (−1 − 1.73i)13-s + (2.31 + 0.543i)15-s + (−0.5 − 0.866i)16-s + (2.18 + 3.78i)17-s + (0.186 + 2.99i)18-s + (−2.5 + 4.33i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.973 − 0.228i)3-s + (−0.249 + 0.433i)4-s − 0.613·5-s + (−0.204 − 0.677i)6-s − 0.353·8-s + (0.895 + 0.445i)9-s + (−0.216 − 0.375i)10-s + 1.31·11-s + (0.342 − 0.364i)12-s + (−0.277 − 0.480i)13-s + (0.597 + 0.140i)15-s + (−0.125 − 0.216i)16-s + (0.530 + 0.918i)17-s + (0.0438 + 0.705i)18-s + (−0.573 + 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.971 - 0.236i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.971 - 0.236i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0714549 + 0.594606i\)
\(L(\frac12)\) \(\approx\) \(0.0714549 + 0.594606i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.37T + 5T^{2} \)
11 \( 1 - 4.37T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.18 - 3.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.37T + 23T^{2} \)
29 \( 1 + (1.37 - 2.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.18 + 8.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.55 - 7.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.37 + 2.37i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.55 - 6.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.05 - 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.55 - 13.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + (-2.55 - 4.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.05 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.74 + 4.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.62 - 2.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.55 - 7.89i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.50340793837842913991305289009, −9.855121571651042560855612634952, −8.531290455818742730260656830760, −7.83876281104667878124101339770, −6.97362593833939523540628652736, −6.12706998132374100489403724241, −5.53418972260422996179318487716, −4.22904794972117482123567903988, −3.72321739260956098462027395558, −1.60944633102318667180547365507, 0.29680947834455573733651518480, 1.86701289254042232053151519910, 3.58691966846064477060217541732, 4.27488739828705691199349531519, 5.11487609696476299226968965753, 6.24708437463745515654714362060, 6.89817361803044741509482742612, 8.042711983809419214745409620229, 9.372317929216188366947963236817, 9.738933598027065595387248594072

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