↑

Properties

Label 2-80-20.7-c5-0-14
Degree $2$
Conductor $80$
Sign $-0.850 - 0.525i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−13.9 − 13.9i)3-s + (−25 − 50i)5-s + (125. − 125. i)7-s + 147i·9-s − 418. i·11-s + (−695 + 695i)13-s + (−349. + 1.04e3i)15-s + (−95 − 95i)17-s − 837.·19-s − 3.50e3·21-s + (2.55e3 + 2.55e3i)23-s + (−1.87e3 + 2.50e3i)25-s + (−1.34e3 + 1.34e3i)27-s + 2.87e3i·29-s − 9.63e3i·31-s + ⋯
L(s)  = 1  + (−0.895 − 0.895i)3-s + (−0.447 − 0.894i)5-s + (0.969 − 0.969i)7-s + 0.604i·9-s − 1.04i·11-s + (−1.14 + 1.14i)13-s + (−0.400 + 1.20i)15-s + (−0.0797 − 0.0797i)17-s − 0.532·19-s − 1.73·21-s + (1.00 + 1.00i)23-s + (−0.600 + 0.800i)25-s + (−0.353 + 0.353i)27-s + 0.635i·29-s − 1.80i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.850 - 0.525i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.173285 + 0.609991i\)
\(L(\frac12)\) \(\approx\) \(0.173285 + 0.609991i\)
\(L(\frac{7}{2})\) 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 \( 1 + (25 + 50i)T \)
good3 \( 1 + (13.9 + 13.9i)T + 243iT^{2} \)
7 \( 1 + (-125. + 125. i)T - 1.68e4iT^{2} \)
11 \( 1 + 418. iT - 1.61e5T^{2} \)
13 \( 1 + (695 - 695i)T - 3.71e5iT^{2} \)
17 \( 1 + (95 + 95i)T + 1.41e6iT^{2} \)
19 \( 1 + 837.T + 2.47e6T^{2} \)
23 \( 1 + (-2.55e3 - 2.55e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 2.87e3iT - 2.05e7T^{2} \)
31 \( 1 + 9.63e3iT - 2.86e7T^{2} \)
37 \( 1 + (5.15e3 + 5.15e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 9.21e3T + 1.15e8T^{2} \)
43 \( 1 + (-1.55e3 - 1.55e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (1.26e4 - 1.26e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-1.13e4 + 1.13e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 837.T + 7.14e8T^{2} \)
61 \( 1 - 1.86e4T + 8.44e8T^{2} \)
67 \( 1 + (-1.73e4 + 1.73e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 5.82e4iT - 1.80e9T^{2} \)
73 \( 1 + (2.40e4 - 2.40e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 9.38e4T + 3.07e9T^{2} \)
83 \( 1 + (3.56e4 + 3.56e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 7.42e4iT - 5.58e9T^{2} \)
97 \( 1 + (4.92e4 + 4.92e4i)T + 8.58e9iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.69433101253789349350639490118, −11.55913302056258149648168489528, −11.13966526871242644667170115790, −9.257090988970116451057025355208, −7.86945230861765453108992579788, −6.99491443650863416297442032977, −5.44974217883828300017346773975, −4.25112417376278024614329650764, −1.47294453288905145997585829670, −0.31255548954118273018838958049, 2.54963076510183620081683577310, 4.55263719669582011396006231868, 5.40260139447437979538996356959, 7.01279501566513600636948915139, 8.364739801927713419480785493789, 10.02857372627348567369575698480, 10.70957784797521593858406950527, 11.73017902450898059621230264741, 12.54448293629289390031679169461, 14.64623698436161490067533967727

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