↑

Properties

Label 2-80-5.4-c19-0-40
Degree $2$
Conductor $80$
Sign $-0.999 - 0.0399i$
Analytic cond. $183.053$
Root an. cond. $13.5297$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4.14e4i·3-s + (−4.36e6 − 1.74e5i)5-s − 4.72e7i·7-s − 5.53e8·9-s + 2.05e9·11-s + 5.51e10i·13-s + (−7.22e9 + 1.80e11i)15-s + 5.78e11i·17-s + 6.40e11·19-s − 1.95e12·21-s − 9.37e12i·23-s + (1.90e13 + 1.52e12i)25-s − 2.52e13i·27-s − 7.47e13·29-s + 2.06e14·31-s + ⋯
L(s)  = 1  − 1.21i·3-s + (−0.999 − 0.0399i)5-s − 0.442i·7-s − 0.476·9-s + 0.263·11-s + 1.44i·13-s + (−0.0485 + 1.21i)15-s + 1.18i·17-s + 0.455·19-s − 0.537·21-s − 1.08i·23-s + (0.996 + 0.0798i)25-s − 0.636i·27-s − 0.957·29-s + 1.40·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0399i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.999 - 0.0399i$
Analytic conductor: \(183.053\)
Root analytic conductor: \(13.5297\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :19/2),\ -0.999 - 0.0399i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.9196980269\)
\(L(\frac12)\) \(\approx\) \(0.9196980269\)
\(L(\frac{21}{2})\) 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 \( 1 + (4.36e6 + 1.74e5i)T \)
good3 \( 1 + 4.14e4iT - 1.16e9T^{2} \)
7 \( 1 + 4.72e7iT - 1.13e16T^{2} \)
11 \( 1 - 2.05e9T + 6.11e19T^{2} \)
13 \( 1 - 5.51e10iT - 1.46e21T^{2} \)
17 \( 1 - 5.78e11iT - 2.39e23T^{2} \)
19 \( 1 - 6.40e11T + 1.97e24T^{2} \)
23 \( 1 + 9.37e12iT - 7.46e25T^{2} \)
29 \( 1 + 7.47e13T + 6.10e27T^{2} \)
31 \( 1 - 2.06e14T + 2.16e28T^{2} \)
37 \( 1 + 3.85e14iT - 6.24e29T^{2} \)
41 \( 1 + 2.89e15T + 4.39e30T^{2} \)
43 \( 1 + 3.27e14iT - 1.08e31T^{2} \)
47 \( 1 + 5.69e15iT - 5.88e31T^{2} \)
53 \( 1 - 5.87e15iT - 5.77e32T^{2} \)
59 \( 1 + 7.38e16T + 4.42e33T^{2} \)
61 \( 1 - 1.34e17T + 8.34e33T^{2} \)
67 \( 1 + 1.67e17iT - 4.95e34T^{2} \)
71 \( 1 + 5.28e17T + 1.49e35T^{2} \)
73 \( 1 + 4.64e17iT - 2.53e35T^{2} \)
79 \( 1 - 1.26e18T + 1.13e36T^{2} \)
83 \( 1 + 6.29e17iT - 2.90e36T^{2} \)
89 \( 1 - 1.81e18T + 1.09e37T^{2} \)
97 \( 1 + 3.16e18iT - 5.60e37T^{2} \)
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

−10.43130823326272969050355889812, −8.866586529038311625495872516023, −7.927054218894132084274878904856, −7.02550970190343291007936202003, −6.33922354717170833398901424773, −4.54841342316795873893939958913, −3.66118134635351784145970400205, −2.12234740959147103316165144785, −1.19900167936341962230986474950, −0.21173538653716047718852068971, 0.969736124864050720438963360221, 2.89650745091832360838172278161, 3.58528886681112337460963546275, 4.72368659687592851082771703771, 5.52161408409614512174989453768, 7.19398166464708167973894095931, 8.219747599685022818690614467220, 9.347456061763627073500966458581, 10.21179926553885114395165654903, 11.29099834687649215732316674944

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