Properties

Label 2-320-80.43-c1-0-5
Degree $2$
Conductor $320$
Sign $0.834 + 0.551i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
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  − 1.28i·3-s + (−0.841 + 2.07i)5-s + (1.13 − 1.13i)7-s + 1.35·9-s + (2.32 − 2.32i)11-s + 1.36·13-s + (2.65 + 1.07i)15-s + (5.25 − 5.25i)17-s + (−3.69 + 3.69i)19-s + (−1.46 − 1.46i)21-s + (0.911 + 0.911i)23-s + (−3.58 − 3.48i)25-s − 5.58i·27-s + (2.37 + 2.37i)29-s − 0.242i·31-s + ⋯
L(s)  = 1  − 0.739i·3-s + (−0.376 + 0.926i)5-s + (0.430 − 0.430i)7-s + 0.452·9-s + (0.700 − 0.700i)11-s + 0.378·13-s + (0.685 + 0.278i)15-s + (1.27 − 1.27i)17-s + (−0.848 + 0.848i)19-s + (−0.318 − 0.318i)21-s + (0.189 + 0.189i)23-s + (−0.716 − 0.697i)25-s − 1.07i·27-s + (0.440 + 0.440i)29-s − 0.0435i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.834 + 0.551i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.834 + 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34051 - 0.402981i\)
\(L(\frac12)\) \(\approx\) \(1.34051 - 0.402981i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.841 - 2.07i)T \)
good3 \( 1 + 1.28iT - 3T^{2} \)
7 \( 1 + (-1.13 + 1.13i)T - 7iT^{2} \)
11 \( 1 + (-2.32 + 2.32i)T - 11iT^{2} \)
13 \( 1 - 1.36T + 13T^{2} \)
17 \( 1 + (-5.25 + 5.25i)T - 17iT^{2} \)
19 \( 1 + (3.69 - 3.69i)T - 19iT^{2} \)
23 \( 1 + (-0.911 - 0.911i)T + 23iT^{2} \)
29 \( 1 + (-2.37 - 2.37i)T + 29iT^{2} \)
31 \( 1 + 0.242iT - 31T^{2} \)
37 \( 1 + 3.34T + 37T^{2} \)
41 \( 1 - 2.66iT - 41T^{2} \)
43 \( 1 + 9.04T + 43T^{2} \)
47 \( 1 + (-7.87 - 7.87i)T + 47iT^{2} \)
53 \( 1 + 5.80iT - 53T^{2} \)
59 \( 1 + (-5.91 - 5.91i)T + 59iT^{2} \)
61 \( 1 + (6.67 - 6.67i)T - 61iT^{2} \)
67 \( 1 - 4.54T + 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 + (-1.49 + 1.49i)T - 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 3.26iT - 83T^{2} \)
89 \( 1 + 9.77T + 89T^{2} \)
97 \( 1 + (1.63 - 1.63i)T - 97iT^{2} \)
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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

−11.64649028420525222026982098798, −10.70419153600495277580812430352, −9.866244120292635939966180809837, −8.460457106170876664108414927300, −7.55427859996449031964227169604, −6.88272436303781289036452394578, −5.88202476306725854632930426864, −4.23659238742724673596266985091, −3.08085591973245328580350741201, −1.30892068777118519883645374739, 1.60627478181651403148048931570, 3.76194752399810663280862524645, 4.54020530799525008581006734902, 5.52065235658445615267057529417, 6.91596160512386181211652338240, 8.213863523319233139150306385185, 8.879984701323471552449014948237, 9.846400092579504080982925119968, 10.68307511198096207405343569782, 11.85066662812480204558027483122

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