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Properties

Label 2-160-5.4-c1-0-3
Degree $2$
Conductor $160$
Sign $0.894 + 0.447i$
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 − 2i)5-s + 3·9-s − 4i·13-s + 8i·17-s + (−3 − 4i)25-s − 10·29-s + 12i·37-s − 10·41-s + (3 − 6i)45-s + 7·49-s + 4i·53-s + 10·61-s + (−8 − 4i)65-s − 16i·73-s + 9·81-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)5-s + 9-s − 1.10i·13-s + 1.94i·17-s + (−0.600 − 0.800i)25-s − 1.85·29-s + 1.97i·37-s − 1.56·41-s + (0.447 − 0.894i)45-s + 49-s + 0.549i·53-s + 1.28·61-s + (−0.992 − 0.496i)65-s − 1.87i·73-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(1.27760\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20672 - 0.284869i\)
\(L(\frac12)\) \(\approx\) \(1.20672 - 0.284869i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
good3 \( 1 - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 8iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 12iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−12.99135949620404659688672522647, −12.07543356892081038834846161312, −10.56903725538474578921268476792, −9.893153518764802002543209859693, −8.671209308200170326934608018083, −7.75263392663315316171910690580, −6.25934352275442499995645379753, −5.14698111002753304188953804645, −3.81810438310669041777406398309, −1.61473517356191733655870362226, 2.16476520096918511412058198152, 3.85170305742133163906454997877, 5.35231253026681953104433630583, 6.86219003417993493021307753474, 7.34659225820132395138253365593, 9.191948760713857735786646205932, 9.836389702397532262256903701307, 11.00391439577334505958100210466, 11.82256781444945886246706336516, 13.13387627056618102443435959454

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