Properties

Degree 2
Conductor $ 2^{5} \cdot 5 $
Sign $-0.960 - 0.279i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.9 − 10.9i)3-s + (−14.9 − 53.8i)5-s + (−75.2 − 75.2i)7-s + 2.76i·9-s − 207. i·11-s + (−233. − 233. i)13-s + (−753. − 426. i)15-s + (−721. + 721. i)17-s + 114.·19-s − 1.64e3·21-s + (−957. + 957. i)23-s + (−2.67e3 + 1.60e3i)25-s + (2.69e3 + 2.69e3i)27-s − 4.18e3i·29-s + 2.74e3i·31-s + ⋯
L(s)  = 1  + (0.703 − 0.703i)3-s + (−0.266 − 0.963i)5-s + (−0.580 − 0.580i)7-s + 0.0113i·9-s − 0.517i·11-s + (−0.383 − 0.383i)13-s + (−0.865 − 0.489i)15-s + (−0.605 + 0.605i)17-s + 0.0724·19-s − 0.816·21-s + (−0.377 + 0.377i)23-s + (−0.857 + 0.514i)25-s + (0.711 + 0.711i)27-s − 0.924i·29-s + 0.512i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.279i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(160\)    =    \(2^{5} \cdot 5\)
\( \varepsilon \)  =  $-0.960 - 0.279i$
motivic weight  =  \(5\)
character  :  $\chi_{160} (63, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 160,\ (\ :5/2),\ -0.960 - 0.279i)\)
\(L(3)\)  \(\approx\)  \(0.9695445655\)
\(L(\frac12)\)  \(\approx\)  \(0.9695445655\)
\(L(\frac{7}{2})\)   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 \( 1 + (14.9 + 53.8i)T \)
good3 \( 1 + (-10.9 + 10.9i)T - 243iT^{2} \)
7 \( 1 + (75.2 + 75.2i)T + 1.68e4iT^{2} \)
11 \( 1 + 207. iT - 1.61e5T^{2} \)
13 \( 1 + (233. + 233. i)T + 3.71e5iT^{2} \)
17 \( 1 + (721. - 721. i)T - 1.41e6iT^{2} \)
19 \( 1 - 114.T + 2.47e6T^{2} \)
23 \( 1 + (957. - 957. i)T - 6.43e6iT^{2} \)
29 \( 1 + 4.18e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.74e3iT - 2.86e7T^{2} \)
37 \( 1 + (-6.90e3 + 6.90e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 8.45e3T + 1.15e8T^{2} \)
43 \( 1 + (1.33e4 - 1.33e4i)T - 1.47e8iT^{2} \)
47 \( 1 + (-6.67e3 - 6.67e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (1.30e4 + 1.30e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 1.13e4T + 7.14e8T^{2} \)
61 \( 1 + 5.60e4T + 8.44e8T^{2} \)
67 \( 1 + (-3.58e4 - 3.58e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 7.42e4iT - 1.80e9T^{2} \)
73 \( 1 + (5.88e4 + 5.88e4i)T + 2.07e9iT^{2} \)
79 \( 1 + 6.53e4T + 3.07e9T^{2} \)
83 \( 1 + (-3.55e4 + 3.55e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 1.11e4iT - 5.58e9T^{2} \)
97 \( 1 + (-8.66e4 + 8.66e4i)T - 8.58e9iT^{2} \)
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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

−11.61802969011668473891333604392, −10.33629902031952480528107274604, −9.168968441609028098719811201304, −8.203568991928382482672590577865, −7.48393607703627191746581179934, −6.12905275336817621256235669030, −4.63427042280646063561128774825, −3.27065678525560571511018929250, −1.70130479412006016918873077545, −0.27544117574174641306384999406, 2.41371282080863118701772723750, 3.35621173994375149158424917354, 4.57886767588894158737566463644, 6.28781941193550249138414897196, 7.24560538683793774045656157736, 8.631949375576560240111599059636, 9.559173429649627742926210921884, 10.26276359449842327492864477854, 11.51952887685971470828358127383, 12.41272979417319321284050158880

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