↑

Properties

Label 2-880-20.19-c2-0-46
Degree $2$
Conductor $880$
Sign $0.538 + 0.842i$
Analytic cond. $23.9782$
Root an. cond. $4.89676$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.71·3-s + (4.21 − 2.69i)5-s + 3.18·7-s − 6.05·9-s + 3.31i·11-s − 14.7i·13-s + (7.22 − 4.61i)15-s + 13.1i·17-s − 26.0i·19-s + 5.46·21-s + 41.9·23-s + (10.5 − 22.6i)25-s − 25.8·27-s + 7.26·29-s − 34.9i·31-s + ⋯
L(s)  = 1  + 0.571·3-s + (0.842 − 0.538i)5-s + 0.455·7-s − 0.673·9-s + 0.301i·11-s − 1.13i·13-s + (0.481 − 0.307i)15-s + 0.776i·17-s − 1.36i·19-s + 0.260·21-s + 1.82·23-s + (0.420 − 0.907i)25-s − 0.956·27-s + 0.250·29-s − 1.12i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.538 + 0.842i$
Analytic conductor: \(23.9782\)
Root analytic conductor: \(4.89676\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1),\ 0.538 + 0.842i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.639219740\)
\(L(\frac12)\) \(\approx\) \(2.639219740\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-4.21 + 2.69i)T \)
11 \( 1 - 3.31iT \)
good3 \( 1 - 1.71T + 9T^{2} \)
7 \( 1 - 3.18T + 49T^{2} \)
13 \( 1 + 14.7iT - 169T^{2} \)
17 \( 1 - 13.1iT - 289T^{2} \)
19 \( 1 + 26.0iT - 361T^{2} \)
23 \( 1 - 41.9T + 529T^{2} \)
29 \( 1 - 7.26T + 841T^{2} \)
31 \( 1 + 34.9iT - 961T^{2} \)
37 \( 1 + 6.70iT - 1.36e3T^{2} \)
41 \( 1 - 39.4T + 1.68e3T^{2} \)
43 \( 1 + 2.00T + 1.84e3T^{2} \)
47 \( 1 - 26.5T + 2.20e3T^{2} \)
53 \( 1 - 52.0iT - 2.80e3T^{2} \)
59 \( 1 + 58.6iT - 3.48e3T^{2} \)
61 \( 1 + 46.1T + 3.72e3T^{2} \)
67 \( 1 - 20.6T + 4.48e3T^{2} \)
71 \( 1 - 5.52iT - 5.04e3T^{2} \)
73 \( 1 - 38.9iT - 5.32e3T^{2} \)
79 \( 1 + 97.1iT - 6.24e3T^{2} \)
83 \( 1 + 24.7T + 6.88e3T^{2} \)
89 \( 1 - 50.4T + 7.92e3T^{2} \)
97 \( 1 - 149. iT - 9.40e3T^{2} \)
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

−9.570205674576287889182375674223, −8.981960887329031288035580587582, −8.280148690537455432103230328129, −7.42671870846654937361574053263, −6.20937431802737710161704380372, −5.37398180555666032566145351825, −4.57753607336799472360928596945, −3.09844719129354773923441832555, −2.25832733034843631115462500618, −0.842767122619228467098339042622, 1.46436822745081993674772486283, 2.58525093472741281342921478916, 3.43670037769146809897101124854, 4.83033735410269911363533176645, 5.74015501438156148454814898123, 6.67422049635511458378137319998, 7.50752339180440441617172016628, 8.608658777582909730699437669044, 9.138239615785386514732384264382, 9.961425443019140880881367948915

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