↑

Properties

Label 2-882-63.4-c1-0-28
Degree $2$
Conductor $882$
Sign $0.244 + 0.969i$
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 + (1.22 − 1.22i)3-s + (−0.499 − 0.866i)4-s − 1.03·5-s + (0.448 + 1.67i)6-s + 0.999·8-s − 2.99i·9-s + (0.517 − 0.896i)10-s − 0.267·11-s + (−1.67 − 0.448i)12-s + (−0.896 + 1.55i)13-s + (−1.26 + 1.26i)15-s + (−0.5 + 0.866i)16-s + (3.41 − 5.91i)17-s + (2.59 + 1.49i)18-s + (−2.19 − 3.79i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.707 − 0.707i)3-s + (−0.249 − 0.433i)4-s − 0.462·5-s + (0.183 + 0.683i)6-s + 0.353·8-s − 0.999i·9-s + (0.163 − 0.283i)10-s − 0.0807·11-s + (−0.482 − 0.129i)12-s + (−0.248 + 0.430i)13-s + (−0.327 + 0.327i)15-s + (−0.125 + 0.216i)16-s + (0.828 − 1.43i)17-s + (0.612 + 0.353i)18-s + (−0.502 − 0.870i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.244 + 0.969i$
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: \((2,\ 882,\ (\ :1/2),\ 0.244 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.970276 - 0.755867i\)
\(L(\frac12)\) \(\approx\) \(0.970276 - 0.755867i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 + 1.22i)T \)
7 \( 1 \)
good5 \( 1 + 1.03T + 5T^{2} \)
11 \( 1 + 0.267T + 11T^{2} \)
13 \( 1 + (0.896 - 1.55i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.41 + 5.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.19 + 3.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.46T + 23T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.34 + 5.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.73 + 6.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.31 + 7.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.133 + 0.232i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.378 - 0.656i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.46 - 9.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.637 - 1.10i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.31 + 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.23 + 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 + (2.70 - 4.69i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.46 - 7.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.29 + 5.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.07 - 15.7i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.395600851620935689651483818020, −9.191885660870050024307789051883, −8.128529635554838760517941094108, −7.31948232038858680982900323719, −6.98337292085936653965910739024, −5.76925851716189313354011736932, −4.67407142822749613009464291116, −3.43756688500122508925129007623, −2.24245909915345919272838759001, −0.62307597140736083682623520851, 1.64945469836414020202288926637, 3.02335403366565256355248606541, 3.73709079471984003758524085248, 4.67265289944412862857143812113, 5.80770390770609247829067479792, 7.26234497743857409980162528820, 8.253455213720956110376472904430, 8.452863704168098352808685753736, 9.705623633716059076722010175877, 10.20490650623750182263839083969

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