Properties

Degree 2
Conductor $ 2^{5} \cdot 5 $
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

\( d \)  =  \(2\)
\( N \)  =  \(160\)    =    \(2^{5} \cdot 5\)
\( \varepsilon \)  =  $0.960 + 0.277i$
motivic weight  =  \(5\)
character  :  $\chi_{160} (63, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 160,\ (\ :5/2),\ 0.960 + 0.277i)\)
\(L(3)\)  \(\approx\)  \(0.9786008140\)
\(L(\frac12)\)  \(\approx\)  \(0.9786008140\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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