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Properties

Label 2-160-160.3-c1-0-12
Degree $2$
Conductor $160$
Sign $0.603 + 0.797i$
Analytic cond. $1.27760$
Root an. cond. $1.13031$
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  + (−1.36 − 0.375i)2-s + (1.22 − 0.508i)3-s + (1.71 + 1.02i)4-s + (0.868 − 2.06i)5-s + (−1.86 + 0.231i)6-s + 0.810·7-s + (−1.95 − 2.04i)8-s + (−0.874 + 0.874i)9-s + (−1.95 + 2.48i)10-s + (0.353 + 0.853i)11-s + (2.62 + 0.384i)12-s + (3.86 − 1.60i)13-s + (−1.10 − 0.304i)14-s + (0.0190 − 2.96i)15-s + (1.89 + 3.52i)16-s + (−4.37 − 4.37i)17-s + ⋯
L(s)  = 1  + (−0.964 − 0.265i)2-s + (0.708 − 0.293i)3-s + (0.858 + 0.512i)4-s + (0.388 − 0.921i)5-s + (−0.760 + 0.0945i)6-s + 0.306·7-s + (−0.691 − 0.722i)8-s + (−0.291 + 0.291i)9-s + (−0.619 + 0.784i)10-s + (0.106 + 0.257i)11-s + (0.758 + 0.111i)12-s + (1.07 − 0.444i)13-s + (−0.295 − 0.0814i)14-s + (0.00490 − 0.766i)15-s + (0.474 + 0.880i)16-s + (−1.06 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.603 + 0.797i$
Analytic conductor: \(1.27760\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1/2),\ 0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.892393 - 0.444025i\)
\(L(\frac12)\) \(\approx\) \(0.892393 - 0.444025i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.375i)T \)
5 \( 1 + (-0.868 + 2.06i)T \)
good3 \( 1 + (-1.22 + 0.508i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 - 0.810T + 7T^{2} \)
11 \( 1 + (-0.353 - 0.853i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-3.86 + 1.60i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + (4.37 + 4.37i)T + 17iT^{2} \)
19 \( 1 + (-0.285 + 0.690i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 - 5.36T + 23T^{2} \)
29 \( 1 + (1.48 - 3.58i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 9.06iT - 31T^{2} \)
37 \( 1 + (0.870 + 0.360i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.572 - 0.572i)T + 41iT^{2} \)
43 \( 1 + (4.42 - 10.6i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (1.59 - 1.59i)T - 47iT^{2} \)
53 \( 1 + (4.62 + 1.91i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.996 - 2.40i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-5.76 - 2.38i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-2.34 - 5.66i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (1.22 + 1.22i)T + 71iT^{2} \)
73 \( 1 + 11.4iT - 73T^{2} \)
79 \( 1 + 17.0iT - 79T^{2} \)
83 \( 1 + (-3.28 - 7.93i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-7.65 - 7.65i)T + 89iT^{2} \)
97 \( 1 + (2.02 - 2.02i)T - 97iT^{2} \)
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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

−12.86077427979801705338707058448, −11.55219000609980390208813065687, −10.71438358386221252014503183409, −9.286726640112562176027424026834, −8.766891070100187433714892937576, −7.928359545879539337868083183360, −6.67917495360681531432729076208, −5.02852558693968935585083019891, −3.01961856392922436421458065183, −1.50986204065286639155935427582, 2.14929834962259121933265969352, 3.63550900697258175856470295588, 5.92893017312638572178992922757, 6.74775984876694768206552572667, 8.135263480873766791468098793987, 8.905551346525765663127700509147, 9.813063179779628958313178667646, 10.95964439419094182200875701517, 11.46286127401078477823052998811, 13.32848055522511358826802151535

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