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Properties

Label 2-320-80.43-c1-0-1
Degree $2$
Conductor $320$
Sign $-0.997 - 0.0731i$
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  + 2.96i·3-s + (−2.22 − 0.177i)5-s + (0.115 − 0.115i)7-s − 5.79·9-s + (−2.95 + 2.95i)11-s + 1.55·13-s + (0.525 − 6.61i)15-s + (0.299 − 0.299i)17-s + (−2.26 + 2.26i)19-s + (0.341 + 0.341i)21-s + (−4.14 − 4.14i)23-s + (4.93 + 0.790i)25-s − 8.28i·27-s + (−0.289 − 0.289i)29-s + 4.18i·31-s + ⋯
L(s)  = 1  + 1.71i·3-s + (−0.996 − 0.0793i)5-s + (0.0435 − 0.0435i)7-s − 1.93·9-s + (−0.892 + 0.892i)11-s + 0.432·13-s + (0.135 − 1.70i)15-s + (0.0726 − 0.0726i)17-s + (−0.519 + 0.519i)19-s + (0.0744 + 0.0744i)21-s + (−0.864 − 0.864i)23-s + (0.987 + 0.158i)25-s − 1.59i·27-s + (−0.0537 − 0.0537i)29-s + 0.751i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.997 - 0.0731i$
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: \((2,\ 320,\ (\ :1/2),\ -0.997 - 0.0731i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0265111 + 0.723791i\)
\(L(\frac12)\) \(\approx\) \(0.0265111 + 0.723791i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.177i)T \)
good3 \( 1 - 2.96iT - 3T^{2} \)
7 \( 1 + (-0.115 + 0.115i)T - 7iT^{2} \)
11 \( 1 + (2.95 - 2.95i)T - 11iT^{2} \)
13 \( 1 - 1.55T + 13T^{2} \)
17 \( 1 + (-0.299 + 0.299i)T - 17iT^{2} \)
19 \( 1 + (2.26 - 2.26i)T - 19iT^{2} \)
23 \( 1 + (4.14 + 4.14i)T + 23iT^{2} \)
29 \( 1 + (0.289 + 0.289i)T + 29iT^{2} \)
31 \( 1 - 4.18iT - 31T^{2} \)
37 \( 1 - 1.63T + 37T^{2} \)
41 \( 1 - 7.61iT - 41T^{2} \)
43 \( 1 - 6.72T + 43T^{2} \)
47 \( 1 + (-4.38 - 4.38i)T + 47iT^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + (-1.63 - 1.63i)T + 59iT^{2} \)
61 \( 1 + (1.23 - 1.23i)T - 61iT^{2} \)
67 \( 1 + 2.49T + 67T^{2} \)
71 \( 1 + 8.00T + 71T^{2} \)
73 \( 1 + (-1.12 + 1.12i)T - 73iT^{2} \)
79 \( 1 - 3.62T + 79T^{2} \)
83 \( 1 + 1.62iT - 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + (-9.69 + 9.69i)T - 97iT^{2} \)
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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

−11.92887908996059436922640625544, −10.78855667408346305809742725330, −10.44704915048179462134898227930, −9.391150470906226044930009511490, −8.450105674642113096010662134052, −7.57429262103085121600432483491, −5.95500831905324708299636755575, −4.65550813777751560582206883052, −4.19077623288109015973932971551, −2.92096209966786574136417831174, 0.50682339378796597651044811130, 2.30620453978640922696306240830, 3.65958161599354638102204449769, 5.46863192738710942620258761530, 6.46530628136116741901177060196, 7.48816909708404970500650872600, 8.054115530755952139987515661552, 8.837752311676852549025619330166, 10.61770195420733831411535921029, 11.46550633457892912341881256738

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