Properties

Degree $2$
Conductor $80$
Sign $0.549 - 0.835i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.36 + 4.55i)2-s + (−18.2 + 18.2i)3-s + (−9.40 − 30.5i)4-s + (−17.6 − 17.6i)5-s + (−21.7 − 144. i)6-s + 13.9i·7-s + (170. + 60.0i)8-s − 424. i·9-s + (139. − 21.0i)10-s + (−220. − 220. i)11-s + (730. + 386. i)12-s + (433. − 433. i)13-s + (−63.3 − 46.7i)14-s + 645.·15-s + (−847. + 575. i)16-s − 25.9·17-s + ⋯
L(s)  = 1  + (−0.594 + 0.804i)2-s + (−1.17 + 1.17i)3-s + (−0.293 − 0.955i)4-s + (−0.316 − 0.316i)5-s + (−0.246 − 1.63i)6-s + 0.107i·7-s + (0.943 + 0.331i)8-s − 1.74i·9-s + (0.442 − 0.0664i)10-s + (−0.549 − 0.549i)11-s + (1.46 + 0.775i)12-s + (0.711 − 0.711i)13-s + (−0.0863 − 0.0637i)14-s + 0.741·15-s + (−0.827 + 0.561i)16-s − 0.0217·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.549 - 0.835i$
Motivic weight: \(5\)
Character: $\chi_{80} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ 0.549 - 0.835i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.526879 + 0.284203i\)
\(L(\frac12)\) \(\approx\) \(0.526879 + 0.284203i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.36 - 4.55i)T \)
5 \( 1 + (17.6 + 17.6i)T \)
good3 \( 1 + (18.2 - 18.2i)T - 243iT^{2} \)
7 \( 1 - 13.9iT - 1.68e4T^{2} \)
11 \( 1 + (220. + 220. i)T + 1.61e5iT^{2} \)
13 \( 1 + (-433. + 433. i)T - 3.71e5iT^{2} \)
17 \( 1 + 25.9T + 1.41e6T^{2} \)
19 \( 1 + (1.56e3 - 1.56e3i)T - 2.47e6iT^{2} \)
23 \( 1 - 1.49e3iT - 6.43e6T^{2} \)
29 \( 1 + (-324. + 324. i)T - 2.05e7iT^{2} \)
31 \( 1 - 7.90e3T + 2.86e7T^{2} \)
37 \( 1 + (3.03e3 + 3.03e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.75e4iT - 1.15e8T^{2} \)
43 \( 1 + (-1.12e4 - 1.12e4i)T + 1.47e8iT^{2} \)
47 \( 1 - 1.69e4T + 2.29e8T^{2} \)
53 \( 1 + (-1.53e4 - 1.53e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (2.44e4 + 2.44e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (2.89e4 - 2.89e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (-4.27e4 + 4.27e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 4.05e4iT - 1.80e9T^{2} \)
73 \( 1 + 872. iT - 2.07e9T^{2} \)
79 \( 1 - 8.71e4T + 3.07e9T^{2} \)
83 \( 1 + (6.30e4 - 6.30e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 5.13e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.17e5T + 8.58e9T^{2} \)
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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

−13.82505266607450294389251549769, −12.26377956934416533746892023183, −10.85867930421860311020262366608, −10.41938119015863022523559257516, −9.102263391869209817387843952494, −7.964639255917187149443985475914, −6.14486829371796277897052291805, −5.41211844624521882078388757527, −4.10456122588173588781062531291, −0.57693108845442145728265938597, 0.790270007198244723198669428583, 2.32042568726959130899947111211, 4.52827156193810192224032692694, 6.45585612310664059862558975102, 7.36493183979755074646786115174, 8.606800709968852730029763892957, 10.35507301282979909303109747439, 11.17768743558050979008162649533, 11.99349597513179494588191775914, 12.86807943076369521916540803385

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