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Properties

Label 2-880-880.147-c1-0-134
Degree $2$
Conductor $880$
Sign $-0.938 + 0.345i$
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  + (1.15 − 0.817i)2-s + (0.618 + 0.200i)3-s + (0.663 − 1.88i)4-s + (−2.02 − 0.939i)5-s + (0.877 − 0.273i)6-s + (−1.18 + 2.32i)7-s + (−0.776 − 2.71i)8-s + (−2.08 − 1.51i)9-s + (−3.10 + 0.574i)10-s + (2.29 − 2.39i)11-s + (0.789 − 1.03i)12-s + (−2.56 − 1.86i)13-s + (0.534 + 3.65i)14-s + (−1.06 − 0.988i)15-s + (−3.11 − 2.50i)16-s + (−5.65 − 0.895i)17-s + ⋯
L(s)  = 1  + (0.815 − 0.578i)2-s + (0.356 + 0.115i)3-s + (0.331 − 0.943i)4-s + (−0.907 − 0.420i)5-s + (0.358 − 0.111i)6-s + (−0.448 + 0.879i)7-s + (−0.274 − 0.961i)8-s + (−0.695 − 0.504i)9-s + (−0.983 + 0.181i)10-s + (0.691 − 0.722i)11-s + (0.227 − 0.298i)12-s + (−0.710 − 0.516i)13-s + (0.142 + 0.976i)14-s + (−0.275 − 0.255i)15-s + (−0.779 − 0.625i)16-s + (−1.37 − 0.217i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.938 + 0.345i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ -0.938 + 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.260382 - 1.45932i\)
\(L(\frac12)\) \(\approx\) \(0.260382 - 1.45932i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 + 0.817i)T \)
5 \( 1 + (2.02 + 0.939i)T \)
11 \( 1 + (-2.29 + 2.39i)T \)
good3 \( 1 + (-0.618 - 0.200i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (1.18 - 2.32i)T + (-4.11 - 5.66i)T^{2} \)
13 \( 1 + (2.56 + 1.86i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.65 + 0.895i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (-4.21 + 2.14i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 + (1.65 - 1.65i)T - 23iT^{2} \)
29 \( 1 + (0.854 + 0.435i)T + (17.0 + 23.4i)T^{2} \)
31 \( 1 + (-2.42 + 3.34i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.28 + 7.02i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-6.45 - 2.09i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.73T + 43T^{2} \)
47 \( 1 + (1.41 + 2.78i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (4.08 - 5.61i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.05 + 0.540i)T + (34.6 + 47.7i)T^{2} \)
61 \( 1 + (-9.69 - 1.53i)T + (58.0 + 18.8i)T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 + (6.10 - 4.43i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-6.79 + 13.3i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (-4.19 - 3.04i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.97 - 6.84i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 2.99T + 89T^{2} \)
97 \( 1 + (2.76 + 17.4i)T + (-92.2 + 29.9i)T^{2} \)
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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

−9.498838059048704595553801434890, −9.206333901523387824906175619642, −8.260469287758950599890870199733, −7.04451435226685544889697105560, −6.04455460844918655533616986965, −5.25566009709886985044283881677, −4.14851437223440895252803700501, −3.28363060528801750496069285447, −2.46987888814597681081082872686, −0.49093273805433951685076322653, 2.30116826671315812294293045948, 3.43202001082224689721210261930, 4.20325232205423540043754639872, 5.03469183098031367565156409248, 6.56559391239587752004120438977, 6.93948866713439243903864118649, 7.77949555086375269924153033651, 8.494769282908619688689126438360, 9.587181958424156837196658156996, 10.71418286540375212489092470295

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