↑

Properties

Label 2-882-9.7-c1-0-39
Degree $2$
Conductor $882$
Sign $-0.766 - 0.642i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s − 1.73i·3-s + (−0.499 − 0.866i)4-s + (−1.5 − 2.59i)5-s + (−1.49 − 0.866i)6-s − 0.999·8-s − 2.99·9-s − 3·10-s + (3 − 5.19i)11-s + (−1.49 + 0.866i)12-s + (1 + 1.73i)13-s + (−4.5 + 2.59i)15-s + (−0.5 + 0.866i)16-s − 6·17-s + (−1.49 + 2.59i)18-s + 7·19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s − 0.999i·3-s + (−0.249 − 0.433i)4-s + (−0.670 − 1.16i)5-s + (−0.612 − 0.353i)6-s − 0.353·8-s − 0.999·9-s − 0.948·10-s + (0.904 − 1.56i)11-s + (−0.433 + 0.249i)12-s + (0.277 + 0.480i)13-s + (−1.16 + 0.670i)15-s + (−0.125 + 0.216i)16-s − 1.45·17-s + (−0.353 + 0.612i)18-s + 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.766 - 0.642i$
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: \((2,\ 882,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.443725 + 1.21912i\)
\(L(\frac12)\) \(\approx\) \(0.443725 + 1.21912i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
7 \( 1 \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.290633417278735376827883738495, −8.789726955683827053751237734212, −8.175858267806827663603888974826, −6.96945011160415676378517493244, −6.08502832549559124301236003416, −5.14872375829298971321837624691, −4.06336083359052502233310536445, −3.09115944200524700898693665764, −1.54597252019072772210147034697, −0.58195978553675455293024134787, 2.55538912827991486450575441755, 3.71897462971167064660627416686, 4.24839739764351505823086831606, 5.33142328398286281969769102218, 6.42009486056554632692770233050, 7.19623098128942279082882958680, 7.899321443447152261510493070964, 9.117445548242253407093474649899, 9.736000199304687786589764240958, 10.59694363920667051924635489392

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