↑

Properties

Label 2-882-63.38-c1-0-12
Degree $2$
Conductor $882$
Sign $0.992 + 0.125i$
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 + (−1.54 − 0.780i)3-s − 4-s + (0.183 − 0.317i)5-s + (−0.780 + 1.54i)6-s + i·8-s + (1.78 + 2.41i)9-s + (−0.317 − 0.183i)10-s + (−0.579 + 0.334i)11-s + (1.54 + 0.780i)12-s + (0.867 − 0.500i)13-s + (−0.531 + 0.347i)15-s + 16-s + (−2.49 + 4.32i)17-s + (2.41 − 1.78i)18-s + (−5.50 + 3.17i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.892 − 0.450i)3-s − 0.5·4-s + (0.0819 − 0.141i)5-s + (−0.318 + 0.631i)6-s + 0.353i·8-s + (0.593 + 0.804i)9-s + (−0.100 − 0.0579i)10-s + (−0.174 + 0.100i)11-s + (0.446 + 0.225i)12-s + (0.240 − 0.138i)13-s + (−0.137 + 0.0897i)15-s + 0.250·16-s + (−0.605 + 1.04i)17-s + (0.569 − 0.419i)18-s + (−1.26 + 0.729i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.992 + 0.125i$
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: \((2,\ 882,\ (\ :1/2),\ 0.992 + 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.913978 - 0.0577027i\)
\(L(\frac12)\) \(\approx\) \(0.913978 - 0.0577027i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.54 + 0.780i)T \)
7 \( 1 \)
good5 \( 1 + (-0.183 + 0.317i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.579 - 0.334i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.867 + 0.500i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.49 - 4.32i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.50 - 3.17i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.66 - 3.84i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.58 - 0.914i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
37 \( 1 + (-2.58 - 4.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.15 - 3.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.24 + 3.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.32T + 47T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.72T + 59T^{2} \)
61 \( 1 + 4.95iT - 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 5.49iT - 71T^{2} \)
73 \( 1 + (3.52 + 2.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 8.35T + 79T^{2} \)
83 \( 1 + (8.50 - 14.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.35 - 9.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.9 - 8.60i)T + (48.5 + 84.0i)T^{2} \)
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

−10.51019654882877552869569448361, −9.418275796227459792609755526096, −8.498344939527386806020426205938, −7.60217673423944278568618386182, −6.52890048977530897398235273155, −5.72598839278764426027903039074, −4.77399185850985251887245384038, −3.80829580950403543107657356814, −2.28956623355845193632810413868, −1.16284276674626848777552557581, 0.58759975036858138841270606990, 2.75190404982113592357957510526, 4.30529968611359561852502927124, 4.82904576103255677826868622135, 5.87679429098861810698103981533, 6.70787314443041267199819189518, 7.22661430456365683511663808708, 8.745889652615511221747941830039, 9.060858137647131601497132574545, 10.32942500613186329235645502541

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