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Properties

Label 2-80-16.5-c5-0-2
Degree $2$
Conductor $80$
Sign $-0.351 + 0.936i$
Analytic cond. $12.8307$
Root an. cond. $3.58199$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.10 + 4.72i)2-s + (−7.07 + 7.07i)3-s + (−12.7 − 29.3i)4-s + (17.6 + 17.6i)5-s + (−11.5 − 55.4i)6-s + 216. i·7-s + (178. + 30.8i)8-s + 142. i·9-s + (−138. + 28.7i)10-s + (−417. − 417. i)11-s + (297. + 117. i)12-s + (−493. + 493. i)13-s + (−1.02e3 − 673. i)14-s − 250.·15-s + (−699. + 747. i)16-s + 483.·17-s + ⋯
L(s)  = 1  + (−0.548 + 0.836i)2-s + (−0.453 + 0.453i)3-s + (−0.397 − 0.917i)4-s + (0.316 + 0.316i)5-s + (−0.130 − 0.628i)6-s + 1.67i·7-s + (0.985 + 0.170i)8-s + 0.588i·9-s + (−0.437 + 0.0908i)10-s + (−1.03 − 1.03i)11-s + (0.596 + 0.235i)12-s + (−0.810 + 0.810i)13-s + (−1.39 − 0.918i)14-s − 0.287·15-s + (−0.683 + 0.730i)16-s + 0.405·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.351 + 0.936i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.351 + 0.936i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.219610 - 0.317126i\)
\(L(\frac12)\) \(\approx\) \(0.219610 - 0.317126i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.10 - 4.72i)T \)
5 \( 1 + (-17.6 - 17.6i)T \)
good3 \( 1 + (7.07 - 7.07i)T - 243iT^{2} \)
7 \( 1 - 216. iT - 1.68e4T^{2} \)
11 \( 1 + (417. + 417. i)T + 1.61e5iT^{2} \)
13 \( 1 + (493. - 493. i)T - 3.71e5iT^{2} \)
17 \( 1 - 483.T + 1.41e6T^{2} \)
19 \( 1 + (-1.92e3 + 1.92e3i)T - 2.47e6iT^{2} \)
23 \( 1 + 591. iT - 6.43e6T^{2} \)
29 \( 1 + (969. - 969. i)T - 2.05e7iT^{2} \)
31 \( 1 + 6.05e3T + 2.86e7T^{2} \)
37 \( 1 + (4.31e3 + 4.31e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 5.21e3iT - 1.15e8T^{2} \)
43 \( 1 + (9.66e3 + 9.66e3i)T + 1.47e8iT^{2} \)
47 \( 1 + 8.66e3T + 2.29e8T^{2} \)
53 \( 1 + (-2.32e4 - 2.32e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (-2.56e4 - 2.56e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (1.11e4 - 1.11e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (-2.77e4 + 2.77e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 3.34e3iT - 1.80e9T^{2} \)
73 \( 1 - 4.73e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.71e4T + 3.07e9T^{2} \)
83 \( 1 + (6.55e4 - 6.55e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 2.44e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.49e5T + 8.58e9T^{2} \)
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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

−14.40471807173591670356027317245, −13.35202496203157589805655735541, −11.70664295821428090698964769321, −10.69898598819149993154895499048, −9.525086210916531687972727385679, −8.610814781817645837171791296663, −7.23202126577574280700799297002, −5.59132513186739259143874510277, −5.23269072611172332156228580745, −2.41214166393893550224728094386, 0.21222512598530331251689572040, 1.46123250930581415365386657479, 3.50511270915008627167393052594, 5.10651598151290721636765956901, 7.20180213049224701110941803328, 7.83887859368530170212764799435, 9.894997865362126292073677509721, 10.11963899261074954214771143630, 11.57655898315700206333699979085, 12.66806801329024307940564458814

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