↑

Properties

Label 2-882-63.38-c1-0-32
Degree $2$
Conductor $882$
Sign $0.344 + 0.938i$
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  + i·2-s + (−1.52 − 0.816i)3-s − 4-s + (1.82 − 3.15i)5-s + (0.816 − 1.52i)6-s − i·8-s + (1.66 + 2.49i)9-s + (3.15 + 1.82i)10-s + (4.38 − 2.53i)11-s + (1.52 + 0.816i)12-s + (−2.94 + 1.69i)13-s + (−5.35 + 3.33i)15-s + 16-s + (0.774 − 1.34i)17-s + (−2.49 + 1.66i)18-s + (0.707 − 0.408i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.881 − 0.471i)3-s − 0.5·4-s + (0.814 − 1.41i)5-s + (0.333 − 0.623i)6-s − 0.353i·8-s + (0.555 + 0.831i)9-s + (0.997 + 0.576i)10-s + (1.32 − 0.763i)11-s + (0.440 + 0.235i)12-s + (−0.816 + 0.471i)13-s + (−1.38 + 0.860i)15-s + 0.250·16-s + (0.187 − 0.325i)17-s + (−0.587 + 0.393i)18-s + (0.162 − 0.0936i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.344 + 0.938i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.344 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.976984 - 0.682371i\)
\(L(\frac12)\) \(\approx\) \(0.976984 - 0.682371i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (1.52 + 0.816i)T \)
7 \( 1 \)
good5 \( 1 + (-1.82 + 3.15i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.38 + 2.53i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.94 - 1.69i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.774 + 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.707 + 0.408i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.47 - 0.850i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.60 + 2.08i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.16iT - 31T^{2} \)
37 \( 1 + (3.39 + 5.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.01 + 1.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.06 + 5.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.74T + 47T^{2} \)
53 \( 1 + (11.4 + 6.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.17T + 59T^{2} \)
61 \( 1 + 7.25iT - 61T^{2} \)
67 \( 1 - 2.45T + 67T^{2} \)
71 \( 1 - 6.74iT - 71T^{2} \)
73 \( 1 + (3.76 + 2.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (0.768 - 1.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.01 - 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.59 - 3.23i)T + (48.5 + 84.0i)T^{2} \)
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

−9.570368919085692639212421984614, −9.224231020315937158319357345341, −8.288323112527583573036589698653, −7.21878367475506666706344343645, −6.43184673150537816461902309610, −5.56528078599760422135674057329, −5.04828937171043235320992527721, −4.02057153189525897175241990403, −1.83011175330394928889103660264, −0.69052990097262057811256969992, 1.55056690673072782504455708909, 2.85654896204627475616128471910, 3.88116356149633358874616583844, 4.94052282333826760663696051946, 5.99025351863619043900476993944, 6.68163487635321737532673743816, 7.52216157119802267111352229221, 9.227095572169669398347418487997, 9.697888811320038050395487203272, 10.39989534715694664793821680379

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