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Properties

Label 2-160-20.7-c1-0-0
Degree $2$
Conductor $160$
Sign $0.525 - 0.850i$
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 + i)3-s + (1 + 2i)5-s + (−3 + 3i)7-s − i·9-s − 2i·11-s + (3 − 3i)13-s + (−1 + 3i)15-s + (1 + i)17-s + 4·19-s − 6·21-s + (−1 − i)23-s + (−3 + 4i)25-s + (4 − 4i)27-s − 10i·31-s + (2 − 2i)33-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s + (0.447 + 0.894i)5-s + (−1.13 + 1.13i)7-s − 0.333i·9-s − 0.603i·11-s + (0.832 − 0.832i)13-s + (−0.258 + 0.774i)15-s + (0.242 + 0.242i)17-s + 0.917·19-s − 1.30·21-s + (−0.208 − 0.208i)23-s + (−0.600 + 0.800i)25-s + (0.769 − 0.769i)27-s − 1.79i·31-s + (0.348 − 0.348i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(1.27760\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1/2),\ 0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15042 + 0.641404i\)
\(L(\frac12)\) \(\approx\) \(1.15042 + 0.641404i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + (-3 + 3i)T - 13iT^{2} \)
17 \( 1 + (-1 - i)T + 17iT^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 + (1 + i)T + 37iT^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + (-5 - 5i)T + 43iT^{2} \)
47 \( 1 + (3 - 3i)T - 47iT^{2} \)
53 \( 1 + (5 - 5i)T - 53iT^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (1 - i)T - 67iT^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 + (-1 + i)T - 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-5 - 5i)T + 83iT^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 + (3 + 3i)T + 97iT^{2} \)
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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

−13.18474799443083813654811262470, −12.07996834419962387871596008197, −10.88969030397398915925824513429, −9.789307228730164201310574010675, −9.249525268418909476740724116321, −8.054598643025885937760046449933, −6.37348603724595080381248089491, −5.76126620900415616532641932154, −3.52042185307861046332936812944, −2.84266483155671739697665966157, 1.53040055245455860800794730650, 3.46280324179960409198367492059, 4.94063009826392165272069476706, 6.55207174327598380544377695432, 7.42810823995328089741566661987, 8.651028716140333276193354862817, 9.606989755576433417518721615983, 10.50408535884726109693427910114, 12.04741651642471995773160919479, 12.96748194101906668600299611542

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