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Properties

Label 2-160-40.29-c1-0-2
Degree $2$
Conductor $160$
Sign $0.987 + 0.160i$
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  + 1.41·3-s + (1.41 − 1.73i)5-s + 2.44i·7-s − 0.999·9-s − 3.46i·11-s + (2.00 − 2.44i)15-s + 4.89i·17-s + 3.46i·19-s + 3.46i·21-s − 2.44i·23-s + (−0.999 − 4.89i)25-s − 5.65·27-s − 4·31-s − 4.89i·33-s + (4.24 + 3.46i)35-s + ⋯
L(s)  = 1  + 0.816·3-s + (0.632 − 0.774i)5-s + 0.925i·7-s − 0.333·9-s − 1.04i·11-s + (0.516 − 0.632i)15-s + 1.18i·17-s + 0.794i·19-s + 0.755i·21-s − 0.510i·23-s + (−0.199 − 0.979i)25-s − 1.08·27-s − 0.718·31-s − 0.852i·33-s + (0.717 + 0.585i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.987 + 0.160i$
Analytic conductor: \(1.27760\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1/2),\ 0.987 + 0.160i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46786 - 0.118507i\)
\(L(\frac12)\) \(\approx\) \(1.46786 - 0.118507i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 1.73i)T \)
good3 \( 1 - 1.41T + 3T^{2} \)
7 \( 1 - 2.44iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 2.44iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4.24T + 43T^{2} \)
47 \( 1 - 7.34iT - 47T^{2} \)
53 \( 1 - 5.65T + 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 + 3.46iT - 61T^{2} \)
67 \( 1 - 4.24T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−12.91263128160951214659368008086, −12.10126541236893761073302719462, −10.80214363082595556972974455849, −9.530617812293066640554114578445, −8.610490931788288429176969281261, −8.224443468460717456966353242797, −6.18496847811091275157070007980, −5.37306533955832967098114330914, −3.54690223371836310274677125087, −2.05406325523293706674455829453, 2.27577330372620020553451553834, 3.56802123144697703378578358297, 5.20126341299700026547296509883, 6.88400947351338394806531387559, 7.48179346508146734748482155634, 8.987713305828881390722723649643, 9.822767102708738431453399301015, 10.75281146162890974282977055369, 11.85802085269407896740951270716, 13.40578010665427744095130949998

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