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Properties

Label 2-160-20.7-c1-0-3
Degree $2$
Conductor $160$
Sign $0.995 - 0.0898i$
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 + (1 − i)7-s − i·9-s + 6i·11-s + (−1 + i)13-s + (3 − i)15-s + (1 + i)17-s − 4·19-s + 2·21-s + (−5 − 5i)23-s + (−3 − 4i)25-s + (4 − 4i)27-s + 8i·29-s − 2i·31-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s + (0.447 − 0.894i)5-s + (0.377 − 0.377i)7-s − 0.333i·9-s + 1.80i·11-s + (−0.277 + 0.277i)13-s + (0.774 − 0.258i)15-s + (0.242 + 0.242i)17-s − 0.917·19-s + 0.436·21-s + (−1.04 − 1.04i)23-s + (−0.600 − 0.800i)25-s + (0.769 − 0.769i)27-s + 1.48i·29-s − 0.359i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.995 - 0.0898i$
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: \((2,\ 160,\ (\ :1/2),\ 0.995 - 0.0898i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41004 + 0.0634432i\)
\(L(\frac12)\) \(\approx\) \(1.41004 + 0.0634432i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 - 6iT - 11T^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 + (-1 - i)T + 17iT^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (5 + 5i)T + 23iT^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 + (7 - 7i)T - 47iT^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (-7 + 7i)T - 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (-9 + 9i)T - 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-5 - 5i)T + 83iT^{2} \)
89 \( 1 - 89T^{2} \)
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

−12.61011343313657734466065240495, −12.34789578347853819677779384122, −10.59254263417348584777203866687, −9.737530630017934070074751495687, −8.994786825222327311658351953573, −7.895872749771244259331481739921, −6.52982603632530620059916845267, −4.85387390315498983689927577944, −4.10282828350672403813017716725, −2.00506365811512203597125664732, 2.14563453935508534164070954675, 3.36002871228799822622667393158, 5.46933778201520942345358131072, 6.48592670710975906264796631890, 7.84145952511783756878884078215, 8.484163872031317758065800093479, 9.887104930470897720256197454554, 10.95012515232120247188036202485, 11.76159118974839900845605201616, 13.25228226055873729293862743578

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