Properties

Degree $2$
Conductor $80$
Sign $-0.581 + 0.813i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (20.3 − 20.3i)3-s + (−46.4 − 31.1i)5-s + (76.9 + 76.9i)7-s − 588. i·9-s − 556. i·11-s + (141. + 141. i)13-s + (−1.58e3 + 312. i)15-s + (−477. + 477. i)17-s − 1.60e3·19-s + 3.13e3·21-s + (346. − 346. i)23-s + (1.19e3 + 2.88e3i)25-s + (−7.03e3 − 7.03e3i)27-s − 7.48e3i·29-s + 7.92e3i·31-s + ⋯
L(s)  = 1  + (1.30 − 1.30i)3-s + (−0.830 − 0.556i)5-s + (0.593 + 0.593i)7-s − 2.41i·9-s − 1.38i·11-s + (0.231 + 0.231i)13-s + (−1.81 + 0.359i)15-s + (−0.400 + 0.400i)17-s − 1.02·19-s + 1.55·21-s + (0.136 − 0.136i)23-s + (0.380 + 0.924i)25-s + (−1.85 − 1.85i)27-s − 1.65i·29-s + 1.48i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.581 + 0.813i$
Motivic weight: \(5\)
Character: $\chi_{80} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.581 + 0.813i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.04784 - 2.03553i\)
\(L(\frac12)\) \(\approx\) \(1.04784 - 2.03553i\)
\(L(\frac{7}{2})\) 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 \( 1 + (46.4 + 31.1i)T \)
good3 \( 1 + (-20.3 + 20.3i)T - 243iT^{2} \)
7 \( 1 + (-76.9 - 76.9i)T + 1.68e4iT^{2} \)
11 \( 1 + 556. iT - 1.61e5T^{2} \)
13 \( 1 + (-141. - 141. i)T + 3.71e5iT^{2} \)
17 \( 1 + (477. - 477. i)T - 1.41e6iT^{2} \)
19 \( 1 + 1.60e3T + 2.47e6T^{2} \)
23 \( 1 + (-346. + 346. i)T - 6.43e6iT^{2} \)
29 \( 1 + 7.48e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.92e3iT - 2.86e7T^{2} \)
37 \( 1 + (-3.32e3 + 3.32e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.87e4T + 1.15e8T^{2} \)
43 \( 1 + (-8.25e3 + 8.25e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (-5.09e3 - 5.09e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (-1.94e4 - 1.94e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 108.T + 7.14e8T^{2} \)
61 \( 1 - 1.42e4T + 8.44e8T^{2} \)
67 \( 1 + (-2.89e4 - 2.89e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 982. iT - 1.80e9T^{2} \)
73 \( 1 + (2.15e4 + 2.15e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 9.38e3T + 3.07e9T^{2} \)
83 \( 1 + (9.45e3 - 9.45e3i)T - 3.93e9iT^{2} \)
89 \( 1 - 8.48e3iT - 5.58e9T^{2} \)
97 \( 1 + (-1.22e5 + 1.22e5i)T - 8.58e9iT^{2} \)
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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

−13.06432440905770726566286535471, −12.21465028574860771426039396794, −11.17782463503980968597477097836, −8.823430000939886900540510805789, −8.555762588687192347245163351584, −7.55668049530340336982832829289, −6.10512970579450500453458594354, −3.93205366615550259006113717626, −2.41472905427548108092111930753, −0.868767072660102397329778793329, 2.45270932408861420428421830277, 3.94121368514735196701443198747, 4.63902760207654684658125113893, 7.26618235554406992297635458881, 8.142916037404379424030031729037, 9.336722437049851539801693247715, 10.42973134224036858882389412601, 11.17381697650353837037112710656, 12.91634258368680686017742586586, 14.28566204090752030178819563145

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