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Properties

Label 2-80-80.29-c5-0-15
Degree $2$
Conductor $80$
Sign $0.341 - 0.939i$
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  + (−3.62 + 4.34i)2-s + (−3.49 + 3.49i)3-s + (−5.71 − 31.4i)4-s + (−25.3 − 49.8i)5-s + (−2.50 − 27.8i)6-s − 221.·7-s + (157. + 89.3i)8-s + 218. i·9-s + (308. + 70.5i)10-s + (337. − 337. i)11-s + (129. + 89.9i)12-s + (458. − 458. i)13-s + (804. − 963. i)14-s + (262. + 85.4i)15-s + (−958. + 359. i)16-s + 2.27e3i·17-s + ⋯
L(s)  = 1  + (−0.640 + 0.767i)2-s + (−0.223 + 0.223i)3-s + (−0.178 − 0.983i)4-s + (−0.453 − 0.891i)5-s + (−0.0283 − 0.315i)6-s − 1.71·7-s + (0.869 + 0.493i)8-s + 0.899i·9-s + (0.974 + 0.223i)10-s + (0.841 − 0.841i)11-s + (0.260 + 0.180i)12-s + (0.752 − 0.752i)13-s + (1.09 − 1.31i)14-s + (0.301 + 0.0980i)15-s + (−0.936 + 0.351i)16-s + 1.90i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.341 - 0.939i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ 0.341 - 0.939i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.638201 + 0.447075i\)
\(L(\frac12)\) \(\approx\) \(0.638201 + 0.447075i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.62 - 4.34i)T \)
5 \( 1 + (25.3 + 49.8i)T \)
good3 \( 1 + (3.49 - 3.49i)T - 243iT^{2} \)
7 \( 1 + 221.T + 1.68e4T^{2} \)
11 \( 1 + (-337. + 337. i)T - 1.61e5iT^{2} \)
13 \( 1 + (-458. + 458. i)T - 3.71e5iT^{2} \)
17 \( 1 - 2.27e3iT - 1.41e6T^{2} \)
19 \( 1 + (-298. - 298. i)T + 2.47e6iT^{2} \)
23 \( 1 - 3.31e3T + 6.43e6T^{2} \)
29 \( 1 + (423. + 423. i)T + 2.05e7iT^{2} \)
31 \( 1 - 770.T + 2.86e7T^{2} \)
37 \( 1 + (-4.08e3 - 4.08e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 3.54e3iT - 1.15e8T^{2} \)
43 \( 1 + (-4.21e3 - 4.21e3i)T + 1.47e8iT^{2} \)
47 \( 1 + 1.68e4iT - 2.29e8T^{2} \)
53 \( 1 + (-785. - 785. i)T + 4.18e8iT^{2} \)
59 \( 1 + (-298. + 298. i)T - 7.14e8iT^{2} \)
61 \( 1 + (-3.18e4 - 3.18e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (-4.16e4 + 4.16e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 3.00e4iT - 1.80e9T^{2} \)
73 \( 1 - 6.02e4T + 2.07e9T^{2} \)
79 \( 1 - 2.75e3T + 3.07e9T^{2} \)
83 \( 1 + (1.53e4 - 1.53e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 7.65e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.38e5iT - 8.58e9T^{2} \)
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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

−13.47916004508660091762032592878, −12.83858011781491675299747157348, −11.12845113053611101928885441677, −10.09790827985571388102848617680, −8.930135088947784609234775862697, −8.102014177425034189872699423881, −6.47868666849691445142833561430, −5.54653736975095810180102153541, −3.78802235147608916968134437465, −0.917647440831466959940640541092, 0.60984434574736259678798299018, 2.86617797794408854476245442455, 3.88371238502773700227451515731, 6.71090823813968011339092689347, 7.05477550448024783979322410465, 9.254328353048469200350927319420, 9.606117054213132981197235052994, 11.13736012068007693822353535789, 11.93872636766997180631368888333, 12.83698618672919567633060359391

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