↑

Properties

Label 2-882-441.142-c1-0-17
Degree $2$
Conductor $882$
Sign $0.575 - 0.817i$
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  + (0.365 − 0.930i)2-s + (−1.32 + 1.11i)3-s + (−0.733 − 0.680i)4-s + (0.491 − 0.236i)5-s + (0.556 + 1.64i)6-s + (−0.0331 + 2.64i)7-s + (−0.900 + 0.433i)8-s + (0.502 − 2.95i)9-s + (−0.0407 − 0.544i)10-s + (1.02 + 1.28i)11-s + (1.73 + 0.0808i)12-s + (1.73 − 4.42i)13-s + (2.45 + 0.997i)14-s + (−0.385 + 0.862i)15-s + (0.0747 + 0.997i)16-s + (−1.63 + 1.51i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.658i)2-s + (−0.764 + 0.645i)3-s + (−0.366 − 0.340i)4-s + (0.219 − 0.105i)5-s + (0.227 + 0.669i)6-s + (−0.0125 + 0.999i)7-s + (−0.318 + 0.153i)8-s + (0.167 − 0.985i)9-s + (−0.0128 − 0.172i)10-s + (0.308 + 0.386i)11-s + (0.499 + 0.0233i)12-s + (0.481 − 1.22i)13-s + (0.654 + 0.266i)14-s + (−0.0996 + 0.222i)15-s + (0.0186 + 0.249i)16-s + (−0.396 + 0.367i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.575 - 0.817i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.575 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02446 + 0.531510i\)
\(L(\frac12)\) \(\approx\) \(1.02446 + 0.531510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.365 + 0.930i)T \)
3 \( 1 + (1.32 - 1.11i)T \)
7 \( 1 + (0.0331 - 2.64i)T \)
good5 \( 1 + (-0.491 + 0.236i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-1.02 - 1.28i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.73 + 4.42i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (1.63 - 1.51i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (2.07 - 3.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.647 - 2.83i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-5.81 - 1.79i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (2.31 - 4.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.69 - 0.830i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-7.92 - 5.40i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-2.96 + 2.02i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (4.62 - 11.7i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-6.12 + 1.88i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (1.01 - 0.694i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (7.33 - 6.80i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-3.74 + 6.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.407 + 1.78i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.33 - 0.502i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (3.26 + 5.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.21 - 15.8i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (3.73 + 9.51i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-1.10 + 1.90i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.39558286884521585450518934337, −9.583476067683884037454546073851, −8.932104581073790397014976687346, −7.906958011391295061074793067679, −6.32697892572160893337212318552, −5.78414646727895641961302623513, −4.97935059210199915316382894005, −3.94613745661541990574417577093, −2.91884188775981701571944023166, −1.41306233095895849671089343357, 0.60528067685916792796737716238, 2.25359833530106170654163936403, 4.04766581266525895836637055537, 4.66093717607596785817353845940, 5.95594274812260146376029197313, 6.58566591075132038688439535449, 7.12679855103893312639848079420, 8.105031074095100044974830428720, 9.012134331207421619474218368958, 10.08369970686501727359021489681

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