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Properties

Label 2-320-80.43-c1-0-7
Degree $2$
Conductor $320$
Sign $0.987 - 0.160i$
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  + 2i·3-s + (1 − 2i)5-s + (3 − 3i)7-s − 9-s + (1 − i)11-s − 2·13-s + (4 + 2i)15-s + (1 − i)17-s + (−3 + 3i)19-s + (6 + 6i)21-s + (1 + i)23-s + (−3 − 4i)25-s + 4i·27-s + (7 + 7i)29-s + 2i·31-s + ⋯
L(s)  = 1  + 1.15i·3-s + (0.447 − 0.894i)5-s + (1.13 − 1.13i)7-s − 0.333·9-s + (0.301 − 0.301i)11-s − 0.554·13-s + (1.03 + 0.516i)15-s + (0.242 − 0.242i)17-s + (−0.688 + 0.688i)19-s + (1.30 + 1.30i)21-s + (0.208 + 0.208i)23-s + (−0.600 − 0.800i)25-s + 0.769i·27-s + (1.29 + 1.29i)29-s + 0.359i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.987 - 0.160i$
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: \((2,\ 320,\ (\ :1/2),\ 0.987 - 0.160i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55441 + 0.125303i\)
\(L(\frac12)\) \(\approx\) \(1.55441 + 0.125303i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 3T^{2} \)
7 \( 1 + (-3 + 3i)T - 7iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 + (3 - 3i)T - 19iT^{2} \)
23 \( 1 + (-1 - i)T + 23iT^{2} \)
29 \( 1 + (-7 - 7i)T + 29iT^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (7 + 7i)T + 47iT^{2} \)
53 \( 1 + 8iT - 53T^{2} \)
59 \( 1 + (-3 - 3i)T + 59iT^{2} \)
61 \( 1 + (1 - i)T - 61iT^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3 - 3i)T - 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (11 - 11i)T - 97iT^{2} \)
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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

−11.49478479548073334636179847494, −10.44146808139289874391322627518, −10.03828948999061258209431281768, −8.885835115585751563821797233951, −8.123015815808663174689436146854, −6.83672439412938143272034418773, −5.18088978233152992064949977803, −4.70574979518464936507486512757, −3.66745985351084340687865058748, −1.44839225881263002157215345488, 1.80819253553579409390043188520, 2.61823717490455430736711609819, 4.68421099951534354003189335228, 5.95958106222581037048951865545, 6.76307353317260986551227096086, 7.71631869699299992646644216015, 8.570992365075063202755649707221, 9.753678489553911571767561693619, 10.85972639865483693286913601197, 11.81853148565904024045506306947

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