Properties

Degree $2$
Conductor $160$
Sign $0.960 - 0.277i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.4 − 13.4i)3-s + (15.0 − 53.8i)5-s + (−76.4 + 76.4i)7-s + 117. i·9-s + 622. i·11-s + (−293. + 293. i)13-s + (−925. + 521. i)15-s + (−1.15e3 − 1.15e3i)17-s + 2.00e3·19-s + 2.05e3·21-s + (1.39e3 + 1.39e3i)23-s + (−2.67e3 − 1.61e3i)25-s + (−1.68e3 + 1.68e3i)27-s − 305. i·29-s − 2.10e3i·31-s + ⋯
L(s)  = 1  + (−0.861 − 0.861i)3-s + (0.268 − 0.963i)5-s + (−0.589 + 0.589i)7-s + 0.485i·9-s + 1.55i·11-s + (−0.481 + 0.481i)13-s + (−1.06 + 0.598i)15-s + (−0.967 − 0.967i)17-s + 1.27·19-s + 1.01·21-s + (0.548 + 0.548i)23-s + (−0.855 − 0.518i)25-s + (−0.443 + 0.443i)27-s − 0.0675i·29-s − 0.393i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.960 - 0.277i$
Motivic weight: \(5\)
Character: $\chi_{160} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 0.960 - 0.277i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9786008140\)
\(L(\frac12)\) \(\approx\) \(0.9786008140\)
\(L(\frac{7}{2})\) 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 \( 1 + (-15.0 + 53.8i)T \)
good3 \( 1 + (13.4 + 13.4i)T + 243iT^{2} \)
7 \( 1 + (76.4 - 76.4i)T - 1.68e4iT^{2} \)
11 \( 1 - 622. iT - 1.61e5T^{2} \)
13 \( 1 + (293. - 293. i)T - 3.71e5iT^{2} \)
17 \( 1 + (1.15e3 + 1.15e3i)T + 1.41e6iT^{2} \)
19 \( 1 - 2.00e3T + 2.47e6T^{2} \)
23 \( 1 + (-1.39e3 - 1.39e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 305. iT - 2.05e7T^{2} \)
31 \( 1 + 2.10e3iT - 2.86e7T^{2} \)
37 \( 1 + (-9.90e3 - 9.90e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.69e4T + 1.15e8T^{2} \)
43 \( 1 + (2.03e3 + 2.03e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (682. - 682. i)T - 2.29e8iT^{2} \)
53 \( 1 + (-2.10e4 + 2.10e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 1.16e4T + 7.14e8T^{2} \)
61 \( 1 + 1.30e3T + 8.44e8T^{2} \)
67 \( 1 + (3.98e4 - 3.98e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 2.54e4iT - 1.80e9T^{2} \)
73 \( 1 + (-1.03e3 + 1.03e3i)T - 2.07e9iT^{2} \)
79 \( 1 + 1.18e4T + 3.07e9T^{2} \)
83 \( 1 + (-4.51e4 - 4.51e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 1.43e5iT - 5.58e9T^{2} \)
97 \( 1 + (2.43e4 + 2.43e4i)T + 8.58e9iT^{2} \)
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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

−12.05710589317661512099450594191, −11.53687890127420008059797494091, −9.662113975780149760413552153754, −9.272891033793494380059541522810, −7.54937025366068494358929476067, −6.71655081371207418400240391958, −5.54487240099500019290296687982, −4.59391223303572748895709203704, −2.32772902192153018731103182030, −0.976546217881951214578717580638, 0.45600610741081031681833466976, 2.89753591575533337412428868412, 4.04431662949481882812070159768, 5.56744056517739819650152462003, 6.32222839660868980281701767486, 7.58746482695090111059575396209, 9.157575616566461149869816999628, 10.29372868765482382970104446999, 10.79402301252043015901055280816, 11.49900793963599113151962354389

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