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Properties

Label 2-160-20.7-c5-0-12
Degree $2$
Conductor $160$
Sign $-0.472 - 0.881i$
Analytic cond. $25.6614$
Root an. cond. $5.06570$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (17.6 + 17.6i)3-s + (55.7 + 3.44i)5-s + (−52.6 + 52.6i)7-s + 381. i·9-s − 126. i·11-s + (−788. + 788. i)13-s + (924. + 1.04e3i)15-s + (425. + 425. i)17-s + 182.·19-s − 1.86e3·21-s + (846. + 846. i)23-s + (3.10e3 + 384. i)25-s + (−2.44e3 + 2.44e3i)27-s + 6.17e3i·29-s − 7.36e3i·31-s + ⋯
L(s)  = 1  + (1.13 + 1.13i)3-s + (0.998 + 0.0616i)5-s + (−0.406 + 0.406i)7-s + 1.56i·9-s − 0.315i·11-s + (−1.29 + 1.29i)13-s + (1.06 + 1.20i)15-s + (0.357 + 0.357i)17-s + 0.116·19-s − 0.920·21-s + (0.333 + 0.333i)23-s + (0.992 + 0.123i)25-s + (−0.644 + 0.644i)27-s + 1.36i·29-s − 1.37i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.472 - 0.881i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ -0.472 - 0.881i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.955150923\)
\(L(\frac12)\) \(\approx\) \(2.955150923\)
\(L(\frac{7}{2})\) 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 \( 1 + (-55.7 - 3.44i)T \)
good3 \( 1 + (-17.6 - 17.6i)T + 243iT^{2} \)
7 \( 1 + (52.6 - 52.6i)T - 1.68e4iT^{2} \)
11 \( 1 + 126. iT - 1.61e5T^{2} \)
13 \( 1 + (788. - 788. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-425. - 425. i)T + 1.41e6iT^{2} \)
19 \( 1 - 182.T + 2.47e6T^{2} \)
23 \( 1 + (-846. - 846. i)T + 6.43e6iT^{2} \)
29 \( 1 - 6.17e3iT - 2.05e7T^{2} \)
31 \( 1 + 7.36e3iT - 2.86e7T^{2} \)
37 \( 1 + (5.03e3 + 5.03e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.90e4T + 1.15e8T^{2} \)
43 \( 1 + (9.23e3 + 9.23e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (1.85e4 - 1.85e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-6.80e3 + 6.80e3i)T - 4.18e8iT^{2} \)
59 \( 1 - 4.70e4T + 7.14e8T^{2} \)
61 \( 1 - 1.43e4T + 8.44e8T^{2} \)
67 \( 1 + (3.03e4 - 3.03e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 4.39e4iT - 1.80e9T^{2} \)
73 \( 1 + (2.98e4 - 2.98e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 5.41e4T + 3.07e9T^{2} \)
83 \( 1 + (-4.50e3 - 4.50e3i)T + 3.93e9iT^{2} \)
89 \( 1 + 2.92e4iT - 5.58e9T^{2} \)
97 \( 1 + (6.39e4 + 6.39e4i)T + 8.58e9iT^{2} \)
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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

−12.46747405906084938015766270541, −11.02581246133363072805505343979, −9.814892144235040693842487169497, −9.466384976124907893411590184566, −8.617830401978077925633343785437, −7.11627936648834341884094206376, −5.64600226199280981545501434800, −4.44392298468837517045745555801, −3.08891314211525767487143536077, −2.06227356812062481815239264430, 0.821623201903202778975844262354, 2.23456100039570877041808649559, 3.11423448679061594532213310450, 5.16353250115495295961783415221, 6.60692558083701497842262605822, 7.44311367392039084085461778065, 8.416109846586319352170238849587, 9.620700088800580017261659438353, 10.26147227243877567174499913315, 12.10991359486006550761911642884

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