↑

Properties

Label 2-880-880.813-c1-0-75
Degree $2$
Conductor $880$
Sign $0.597 - 0.802i$
Analytic cond. $7.02683$
Root an. cond. $2.65081$
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  + 1.41·2-s + i·3-s + 2.00·4-s + 2.23·5-s + 1.41i·6-s + (−1.58 + 1.58i)7-s + 2.82·8-s + 2·9-s + 3.16·10-s + (−1.52 + 2.94i)11-s + 2.00i·12-s − 4.24·13-s + (−2.23 + 2.23i)14-s + 2.23i·15-s + 4.00·16-s + (3.53 + 3.53i)17-s + ⋯
L(s)  = 1  + 1.00·2-s + 0.577i·3-s + 1.00·4-s + 0.999·5-s + 0.577i·6-s + (−0.597 + 0.597i)7-s + 1.00·8-s + 0.666·9-s + 1.00·10-s + (−0.460 + 0.887i)11-s + 0.577i·12-s − 1.17·13-s + (−0.597 + 0.597i)14-s + 0.577i·15-s + 1.00·16-s + (0.857 + 0.857i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.597 - 0.802i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (813, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.597 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.99413 + 1.50363i\)
\(L(\frac12)\) \(\approx\) \(2.99413 + 1.50363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
5 \( 1 - 2.23T \)
11 \( 1 + (1.52 - 2.94i)T \)
good3 \( 1 - iT - 3T^{2} \)
7 \( 1 + (1.58 - 1.58i)T - 7iT^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + (-3.53 - 3.53i)T + 17iT^{2} \)
19 \( 1 + (5.28 + 5.28i)T + 19iT^{2} \)
23 \( 1 + (-5.23 + 5.23i)T - 23iT^{2} \)
29 \( 1 + (-3.32 - 3.32i)T + 29iT^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 + 8.70iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 7.40iT - 43T^{2} \)
47 \( 1 + (-4.47 + 4.47i)T - 47iT^{2} \)
53 \( 1 - 3.76T + 53T^{2} \)
59 \( 1 + (0.763 + 0.763i)T + 59iT^{2} \)
61 \( 1 + (1.58 - 1.58i)T - 61iT^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 8.23iT - 71T^{2} \)
73 \( 1 + (1.08 + 1.08i)T + 73iT^{2} \)
79 \( 1 + 1.08T + 79T^{2} \)
83 \( 1 - 6.65T + 83T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (2.70 - 2.70i)T - 97iT^{2} \)
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

−10.53736785664457160015797281093, −9.586481993553408742553940185348, −8.846903839132873547267259711454, −7.21776117725527041697756644179, −6.81307556396817656401855378348, −5.59503322459731830823610217938, −5.02284503062065149976658731199, −4.12104257525385065105355338381, −2.76497613023996846311459349143, −2.01423880544415895202362250962, 1.31679769904445719895099435886, 2.53913121458078146230316507591, 3.51851863988019814622298640152, 4.81064222497932049830610310174, 5.66325067401495295426911814984, 6.47720291815818961501702652611, 7.23329780736764507850702284677, 7.920416606111271266469084647448, 9.475737119783982015866084440824, 10.13924172171957868603708241754

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