Properties

Degree $2$
Conductor $80$
Sign $0.983 + 0.178i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14i·3-s + (55 + 10i)5-s + 158i·7-s + 47·9-s + 148·11-s + 684i·13-s + (140 − 770i)15-s − 2.04e3i·17-s + 2.22e3·19-s + 2.21e3·21-s + 1.24e3i·23-s + (2.92e3 + 1.10e3i)25-s − 4.06e3i·27-s + 270·29-s + 2.04e3·31-s + ⋯
L(s)  = 1  − 0.898i·3-s + (0.983 + 0.178i)5-s + 1.21i·7-s + 0.193·9-s + 0.368·11-s + 1.12i·13-s + (0.160 − 0.883i)15-s − 1.71i·17-s + 1.41·19-s + 1.09·21-s + 0.491i·23-s + (0.936 + 0.352i)25-s − 1.07i·27-s + 0.0596·29-s + 0.382·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.983 + 0.178i$
Motivic weight: \(5\)
Character: $\chi_{80} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ 0.983 + 0.178i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.24836 - 0.202734i\)
\(L(\frac12)\) \(\approx\) \(2.24836 - 0.202734i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-55 - 10i)T \)
good3 \( 1 + 14iT - 243T^{2} \)
7 \( 1 - 158iT - 1.68e4T^{2} \)
11 \( 1 - 148T + 1.61e5T^{2} \)
13 \( 1 - 684iT - 3.71e5T^{2} \)
17 \( 1 + 2.04e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.22e3T + 2.47e6T^{2} \)
23 \( 1 - 1.24e3iT - 6.43e6T^{2} \)
29 \( 1 - 270T + 2.05e7T^{2} \)
31 \( 1 - 2.04e3T + 2.86e7T^{2} \)
37 \( 1 - 4.37e3iT - 6.93e7T^{2} \)
41 \( 1 + 2.39e3T + 1.15e8T^{2} \)
43 \( 1 + 2.29e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.06e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.96e3iT - 4.18e8T^{2} \)
59 \( 1 + 3.97e4T + 7.14e8T^{2} \)
61 \( 1 + 4.22e4T + 8.44e8T^{2} \)
67 \( 1 - 3.20e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.24e3T + 1.80e9T^{2} \)
73 \( 1 - 3.01e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.52e4T + 3.07e9T^{2} \)
83 \( 1 - 2.78e4iT - 3.93e9T^{2} \)
89 \( 1 - 8.52e4T + 5.58e9T^{2} \)
97 \( 1 - 9.72e4iT - 8.58e9T^{2} \)
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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

−13.58248784421289701729257868103, −12.19671509682654663964408871948, −11.59846747118026925394831266567, −9.723601451032213785933466091870, −9.051773555324886597155218867365, −7.34926583302124053202932681555, −6.37389826249377757064204605115, −5.12958350681591005540140504081, −2.66908325633065254032997145101, −1.43484698478187625651289244074, 1.22449497840952489148991991128, 3.50316828688008904089271131000, 4.79017797052126808819223439258, 6.16985539287195084585772662105, 7.70784665236488771735719467181, 9.246543034284442442540938365465, 10.27991311618157684912840832220, 10.67475989277613844932529892691, 12.56767908793262022207124289458, 13.50659965957190715628203683561

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