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Properties

Label 2-880-11.3-c1-0-14
Degree $2$
Conductor $880$
Sign $0.594 + 0.803i$
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  + (−0.220 − 0.678i)3-s + (−0.809 − 0.587i)5-s + (−0.116 + 0.357i)7-s + (2.01 − 1.46i)9-s + (0.107 + 3.31i)11-s + (2.28 − 1.66i)13-s + (−0.220 + 0.678i)15-s + (3.91 + 2.84i)17-s + (−0.905 − 2.78i)19-s + 0.268·21-s − 3.77·23-s + (0.309 + 0.951i)25-s + (−3.16 − 2.30i)27-s + (2.60 − 8.03i)29-s + (6.50 − 4.72i)31-s + ⋯
L(s)  = 1  + (−0.127 − 0.391i)3-s + (−0.361 − 0.262i)5-s + (−0.0439 + 0.135i)7-s + (0.671 − 0.488i)9-s + (0.0322 + 0.999i)11-s + (0.634 − 0.461i)13-s + (−0.0569 + 0.175i)15-s + (0.949 + 0.689i)17-s + (−0.207 − 0.639i)19-s + 0.0585·21-s − 0.786·23-s + (0.0618 + 0.190i)25-s + (−0.609 − 0.443i)27-s + (0.484 − 1.49i)29-s + (1.16 − 0.848i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.594 + 0.803i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.594 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33136 - 0.671236i\)
\(L(\frac12)\) \(\approx\) \(1.33136 - 0.671236i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.107 - 3.31i)T \)
good3 \( 1 + (0.220 + 0.678i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.116 - 0.357i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.28 + 1.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.91 - 2.84i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.905 + 2.78i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.77T + 23T^{2} \)
29 \( 1 + (-2.60 + 8.03i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.50 + 4.72i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.877 + 2.70i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.14 + 3.53i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.48T + 43T^{2} \)
47 \( 1 + (0.800 + 2.46i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.0394 + 0.0286i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.509 + 1.56i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.03 + 5.11i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 + (-11.4 - 8.30i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.158 + 0.488i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.5 + 7.63i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.21 - 1.60i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (13.0 - 9.44i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.971029824722937668986533115645, −9.276178089783979870466304099560, −8.092113585684683918531640118580, −7.61468902148330493545822462315, −6.52716256099334092730364907081, −5.81805018001164054838183064518, −4.49988318694834729374633426678, −3.78649485366392744960321214371, −2.26146453554270525046484533887, −0.883685741068940286410924034009, 1.28745732477701713372980737879, 3.01720645596976070288246653134, 3.90698911433499621269502536186, 4.87890066365125238839876061510, 5.91487655173309201527669676728, 6.83719958060996317364170119831, 7.79745009714599433228856908265, 8.517260687939952552868749426071, 9.530025526667447311884818070782, 10.43195351057346627961173933767

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