↑

Properties

Label 2-882-63.4-c1-0-37
Degree $2$
Conductor $882$
Sign $-0.975 - 0.220i$
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 + (1.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 3·5-s − 1.73i·6-s − 0.999·8-s + (1.5 − 2.59i)9-s + (−1.5 + 2.59i)10-s − 3·11-s + (−1.49 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−4.5 + 2.59i)15-s + (−0.5 + 0.866i)16-s + (1.5 − 2.59i)17-s + (−1.5 − 2.59i)18-s + (−3.5 − 6.06i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.866 − 0.499i)3-s + (−0.249 − 0.433i)4-s − 1.34·5-s − 0.707i·6-s − 0.353·8-s + (0.5 − 0.866i)9-s + (−0.474 + 0.821i)10-s − 0.904·11-s + (−0.433 − 0.250i)12-s + (−0.138 + 0.240i)13-s + (−1.16 + 0.670i)15-s + (−0.125 + 0.216i)16-s + (0.363 − 0.630i)17-s + (−0.353 − 0.612i)18-s + (−0.802 − 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.975 - 0.220i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115500 + 1.03478i\)
\(L(\frac12)\) \(\approx\) \(0.115500 + 1.03478i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + 3T + 5T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 9T + 23T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.708241134103259692873776403241, −8.735121365276763786226819668956, −8.008555520496974770775095554190, −7.35905522455404569631532918585, −6.34612927954513809526707694918, −4.86299925282796577590166867486, −4.05717899331588196708266165056, −3.11074172310220503733205894743, −2.18821681019475925448855148512, −0.37947664452942209646127935149, 2.35647720321303614509890025667, 3.77037182893131441837864322630, 3.99550013219839204136959435936, 5.19078692929344181832009996439, 6.26627247333053901676351892004, 7.67725208445889410329675110591, 7.957909151609678218279155514398, 8.433707218054355096952850014232, 9.768264604629468990740190544816, 10.40566914714433396709764231678

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