↑

Properties

Label 2-882-7.4-c3-0-13
Degree $2$
Conductor $882$
Sign $0.991 - 0.126i$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (−3.5 − 6.06i)5-s + 7.99·8-s + (−7 + 12.1i)10-s + (17.5 − 30.3i)11-s − 66·13-s + (−8 − 13.8i)16-s + (−29.5 + 51.0i)17-s + (68.5 + 118. i)19-s + 28·20-s − 70·22-s + (−3.5 − 6.06i)23-s + (38 − 65.8i)25-s + (66 + 114. i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.313 − 0.542i)5-s + 0.353·8-s + (−0.221 + 0.383i)10-s + (0.479 − 0.830i)11-s − 1.40·13-s + (−0.125 − 0.216i)16-s + (−0.420 + 0.728i)17-s + (0.827 + 1.43i)19-s + 0.313·20-s − 0.678·22-s + (−0.0317 − 0.0549i)23-s + (0.303 − 0.526i)25-s + (0.497 + 0.862i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.022900609\)
\(L(\frac12)\) \(\approx\) \(1.022900609\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (3.5 + 6.06i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-17.5 + 30.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 66T + 2.19e3T^{2} \)
17 \( 1 + (29.5 - 51.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-68.5 - 118. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 106T + 2.43e4T^{2} \)
31 \( 1 + (-37.5 + 64.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 498T + 6.89e4T^{2} \)
43 \( 1 - 260T + 7.95e4T^{2} \)
47 \( 1 + (-85.5 - 148. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (208.5 - 361. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-8.5 + 14.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-25.5 - 44.1i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (219.5 - 380. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 784T + 3.57e5T^{2} \)
73 \( 1 + (-147.5 + 255. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-247.5 - 428. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 932T + 5.71e5T^{2} \)
89 \( 1 + (-436.5 - 756. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 290T + 9.12e5T^{2} \)
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

−9.783844137183922726823207024269, −8.997042314847306825319242943007, −8.187452947395834703355436028831, −7.51927821940859444470299721761, −6.30000082153689471637851609023, −5.23451480780146545295883837705, −4.21354458573971250352090719392, −3.32198324117737817605549476503, −2.03693154474428314659159210349, −0.832043311305833913523232234021, 0.40338328084342301524641908582, 2.07252921039306952139782110772, 3.25449409154516855377114838133, 4.67988562440756188989188847818, 5.21205112577994915283732849663, 6.75390374056931334616334687400, 7.07178880730073154365133204922, 7.78204159044455625372466350830, 9.076260019416269867917586082603, 9.520398906676197926588163413478

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