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Properties

Label 2-880-880.197-c1-0-117
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

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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: $\chi_{880} (197, \cdot )$
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 \( 1 + (2.70 + 2.70i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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