↑

Properties

Label 2-882-21.5-c1-0-5
Degree $2$
Conductor $882$
Sign $0.999 + 0.0285i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.358 + 0.621i)5-s − 0.999i·8-s + (0.621 − 0.358i)10-s + (2.59 − 1.5i)11-s + 2.44i·13-s + (−0.5 + 0.866i)16-s + (−2.95 − 5.12i)17-s + (5.12 + 2.95i)19-s − 0.717·20-s − 3·22-s + (−3.67 − 2.12i)23-s + (2.24 + 3.88i)25-s + (1.22 − 2.12i)26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.160 + 0.277i)5-s − 0.353i·8-s + (0.196 − 0.113i)10-s + (0.783 − 0.452i)11-s + 0.679i·13-s + (−0.125 + 0.216i)16-s + (−0.717 − 1.24i)17-s + (1.17 + 0.678i)19-s − 0.160·20-s − 0.639·22-s + (−0.766 − 0.442i)23-s + (0.448 + 0.776i)25-s + (0.240 − 0.416i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0285i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.999 + 0.0285i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16542 - 0.0166118i\)
\(L(\frac12)\) \(\approx\) \(1.16542 - 0.0166118i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.358 - 0.621i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (2.95 + 5.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.12 - 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.67 + 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.24iT - 29T^{2} \)
31 \( 1 + (-7.86 + 4.54i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.121 + 0.210i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + 0.242T + 43T^{2} \)
47 \( 1 + (2.95 - 5.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.27 + 3.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.03 - 6.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.878 + 0.507i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 + (-1.24 + 0.717i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.37 + 2.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.63T + 83T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.5iT - 97T^{2} \)
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

−10.00549538128055183969981614503, −9.313571705693782483087613265571, −8.662939085809014831329246771016, −7.58116353143007913551541015783, −6.90772853517415410480401852979, −5.96276170963510567854847921548, −4.62993750930474617695084002409, −3.57609637638814789604612707075, −2.52317465397626077753882305924, −1.05983981774800557796664473423, 0.921958499226523987839982984909, 2.37428352175997542567851884347, 3.87337227124576847480095321658, 4.87210561872456002430514270402, 6.02251739358501573278288847880, 6.71705452042696099357111012116, 7.78715647126560771656769761031, 8.376245877538508571028464184109, 9.297389975734942590272903998907, 9.982045182737992740675596881247

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