↑

Properties

Label 2-80-16.13-c1-0-5
Degree $2$
Conductor $80$
Sign $0.686 + 0.727i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.09 − 0.889i)2-s + (−0.120 − 0.120i)3-s + (0.418 − 1.95i)4-s + (−0.707 + 0.707i)5-s + (−0.238 − 0.0252i)6-s + 2.66i·7-s + (−1.27 − 2.52i)8-s − 2.97i·9-s + (−0.148 + 1.40i)10-s + (−3.49 + 3.49i)11-s + (−0.284 + 0.184i)12-s + (2.94 + 2.94i)13-s + (2.37 + 2.93i)14-s + 0.169·15-s + (−3.64 − 1.63i)16-s + 1.85·17-s + ⋯
L(s)  = 1  + (0.777 − 0.628i)2-s + (−0.0692 − 0.0692i)3-s + (0.209 − 0.977i)4-s + (−0.316 + 0.316i)5-s + (−0.0974 − 0.0103i)6-s + 1.00i·7-s + (−0.452 − 0.892i)8-s − 0.990i·9-s + (−0.0470 + 0.444i)10-s + (−1.05 + 1.05i)11-s + (−0.0822 + 0.0532i)12-s + (0.815 + 0.815i)13-s + (0.634 + 0.784i)14-s + 0.0438·15-s + (−0.912 − 0.409i)16-s + 0.448·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.686 + 0.727i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.686 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15620 - 0.498769i\)
\(L(\frac12)\) \(\approx\) \(1.15620 - 0.498769i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 + 0.889i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.120 + 0.120i)T + 3iT^{2} \)
7 \( 1 - 2.66iT - 7T^{2} \)
11 \( 1 + (3.49 - 3.49i)T - 11iT^{2} \)
13 \( 1 + (-2.94 - 2.94i)T + 13iT^{2} \)
17 \( 1 - 1.85T + 17T^{2} \)
19 \( 1 + (3.44 + 3.44i)T + 19iT^{2} \)
23 \( 1 + 0.707iT - 23T^{2} \)
29 \( 1 + (3.49 + 3.49i)T + 29iT^{2} \)
31 \( 1 - 6.84T + 31T^{2} \)
37 \( 1 + (0.0975 - 0.0975i)T - 37iT^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (-4.43 + 4.43i)T - 43iT^{2} \)
47 \( 1 + 1.89T + 47T^{2} \)
53 \( 1 + (7.43 - 7.43i)T - 53iT^{2} \)
59 \( 1 + (-0.959 + 0.959i)T - 59iT^{2} \)
61 \( 1 + (-6.49 - 6.49i)T + 61iT^{2} \)
67 \( 1 + (-3.49 - 3.49i)T + 67iT^{2} \)
71 \( 1 - 7.86iT - 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 + 6.70T + 79T^{2} \)
83 \( 1 + (3.87 + 3.87i)T + 83iT^{2} \)
89 \( 1 - 10.5iT - 89T^{2} \)
97 \( 1 - 4.79T + 97T^{2} \)
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

−14.23999436590030247274426277155, −12.92967874858091376482797568087, −12.16718431993555287663477665261, −11.28318028511447054200071947552, −10.02731191576842261792548863641, −8.847625755503298800933982657784, −6.92545453694030598168826981481, −5.72702093116153551512454909827, −4.18279215762973442274404449718, −2.48720399837190436023573471264, 3.38028737164568903032151918570, 4.85044760492588689779600432482, 6.06093940610133685757631829097, 7.79989498170825628106330563327, 8.225908749823123915281804373468, 10.46440698273592368723416579831, 11.25937282014677062778248834614, 12.89971593051123853098059524432, 13.39572566766988987564733095866, 14.39847697888012679555592258747

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