Properties

Degree 2
Conductor $ 2^{5} \cdot 5 $
Sign $-0.692 + 0.721i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−16.5 − 16.5i)3-s + (13.9 + 54.1i)5-s + (−2.15 + 2.15i)7-s + 304. i·9-s − 255. i·11-s + (111. − 111. i)13-s + (665. − 1.12e3i)15-s + (998. + 998. i)17-s + 1.94e3·19-s + 71.4·21-s + (−1.61e3 − 1.61e3i)23-s + (−2.73e3 + 1.50e3i)25-s + (1.02e3 − 1.02e3i)27-s − 5.84e3i·29-s − 1.60e3i·31-s + ⋯
L(s)  = 1  + (−1.06 − 1.06i)3-s + (0.249 + 0.968i)5-s + (−0.0166 + 0.0166i)7-s + 1.25i·9-s − 0.637i·11-s + (0.183 − 0.183i)13-s + (0.763 − 1.29i)15-s + (0.838 + 0.838i)17-s + 1.23·19-s + 0.0353·21-s + (−0.636 − 0.636i)23-s + (−0.875 + 0.482i)25-s + (0.270 − 0.270i)27-s − 1.29i·29-s − 0.300i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(160\)    =    \(2^{5} \cdot 5\)
\( \varepsilon \)  =  $-0.692 + 0.721i$
motivic weight  =  \(5\)
character  :  $\chi_{160} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 160,\ (\ :5/2),\ -0.692 + 0.721i)\)
\(L(3)\)  \(\approx\)  \(0.8207110257\)
\(L(\frac12)\)  \(\approx\)  \(0.8207110257\)
\(L(\frac{7}{2})\)   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 \( 1 + (-13.9 - 54.1i)T \)
good3 \( 1 + (16.5 + 16.5i)T + 243iT^{2} \)
7 \( 1 + (2.15 - 2.15i)T - 1.68e4iT^{2} \)
11 \( 1 + 255. iT - 1.61e5T^{2} \)
13 \( 1 + (-111. + 111. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-998. - 998. i)T + 1.41e6iT^{2} \)
19 \( 1 - 1.94e3T + 2.47e6T^{2} \)
23 \( 1 + (1.61e3 + 1.61e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 5.84e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.60e3iT - 2.86e7T^{2} \)
37 \( 1 + (1.13e4 + 1.13e4i)T + 6.93e7iT^{2} \)
41 \( 1 + 8.43e3T + 1.15e8T^{2} \)
43 \( 1 + (1.36e4 + 1.36e4i)T + 1.47e8iT^{2} \)
47 \( 1 + (1.27e4 - 1.27e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-2.03e3 + 2.03e3i)T - 4.18e8iT^{2} \)
59 \( 1 - 2.75e4T + 7.14e8T^{2} \)
61 \( 1 - 9.59e3T + 8.44e8T^{2} \)
67 \( 1 + (-3.74e4 + 3.74e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 6.00e4iT - 1.80e9T^{2} \)
73 \( 1 + (-2.50e4 + 2.50e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 4.92e4T + 3.07e9T^{2} \)
83 \( 1 + (4.46e4 + 4.46e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 2.04e4iT - 5.58e9T^{2} \)
97 \( 1 + (1.00e5 + 1.00e5i)T + 8.58e9iT^{2} \)
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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

−11.65701447977326675362494946775, −10.82482958441006225699108854265, −9.885099251690917659966855300804, −8.138441513452233258704012206067, −7.18043811076042657125381307798, −6.18428789648197835591935995889, −5.54052492714739461626071642805, −3.44144742089485956520511937842, −1.80642404621072424911334776632, −0.34058295467615935608739459195, 1.27371045554518024638722051706, 3.62105253023422332761657436308, 5.07739126182240708797778714694, 5.30062749588064697001532593026, 6.89940738387721413146518322060, 8.436262994508558100294339321774, 9.800404284530722311425572016679, 9.988220428994553684029683993526, 11.59049925891709325905627809119, 11.95677283288410308122470117440

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