↑

Properties

Label 2-80-5.4-c19-0-5
Degree $2$
Conductor $80$
Sign $-0.929 + 0.369i$
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  + 1.87e4i·3-s + (4.05e6 − 1.61e6i)5-s + 1.79e8i·7-s + 8.10e8·9-s + 3.61e9·11-s + 7.48e9i·13-s + (3.03e10 + 7.60e10i)15-s − 2.32e11i·17-s − 2.12e12·19-s − 3.37e12·21-s + 6.69e12i·23-s + (1.38e13 − 1.31e13i)25-s + 3.69e13i·27-s − 7.26e13·29-s − 2.51e14·31-s + ⋯
L(s)  = 1  + 0.550i·3-s + (0.929 − 0.369i)5-s + 1.68i·7-s + 0.697·9-s + 0.462·11-s + 0.195i·13-s + (0.203 + 0.511i)15-s − 0.474i·17-s − 1.51·19-s − 0.926·21-s + 0.774i·23-s + (0.726 − 0.687i)25-s + 0.933i·27-s − 0.929·29-s − 1.70·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.929 + 0.369i$
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: \((2,\ 80,\ (\ :19/2),\ -0.929 + 0.369i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.9981781022\)
\(L(\frac12)\) \(\approx\) \(0.9981781022\)
\(L(\frac{21}{2})\) 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 \( 1 + (-4.05e6 + 1.61e6i)T \)
good3 \( 1 - 1.87e4iT - 1.16e9T^{2} \)
7 \( 1 - 1.79e8iT - 1.13e16T^{2} \)
11 \( 1 - 3.61e9T + 6.11e19T^{2} \)
13 \( 1 - 7.48e9iT - 1.46e21T^{2} \)
17 \( 1 + 2.32e11iT - 2.39e23T^{2} \)
19 \( 1 + 2.12e12T + 1.97e24T^{2} \)
23 \( 1 - 6.69e12iT - 7.46e25T^{2} \)
29 \( 1 + 7.26e13T + 6.10e27T^{2} \)
31 \( 1 + 2.51e14T + 2.16e28T^{2} \)
37 \( 1 - 2.00e14iT - 6.24e29T^{2} \)
41 \( 1 + 2.32e15T + 4.39e30T^{2} \)
43 \( 1 + 1.85e15iT - 1.08e31T^{2} \)
47 \( 1 + 9.40e15iT - 5.88e31T^{2} \)
53 \( 1 + 1.66e16iT - 5.77e32T^{2} \)
59 \( 1 - 1.59e15T + 4.42e33T^{2} \)
61 \( 1 + 4.33e16T + 8.34e33T^{2} \)
67 \( 1 - 2.54e17iT - 4.95e34T^{2} \)
71 \( 1 - 2.27e17T + 1.49e35T^{2} \)
73 \( 1 - 8.76e17iT - 2.53e35T^{2} \)
79 \( 1 + 5.76e17T + 1.13e36T^{2} \)
83 \( 1 + 5.61e17iT - 2.90e36T^{2} \)
89 \( 1 - 2.36e18T + 1.09e37T^{2} \)
97 \( 1 - 1.90e18iT - 5.60e37T^{2} \)
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

−11.38184811668026856214533228641, −10.08223622553066572727256815123, −9.219856859114993330762873522821, −8.659505471591411829658549483639, −6.88232953310869809112587913096, −5.73338352010481618993683641745, −5.02554925482121889130753218494, −3.72146794707000056442888313122, −2.23893397678725500062172277633, −1.62888878761635680123041231055, 0.16014667035789976890708086466, 1.31895644348986146332979784233, 1.99898456077819246578350356010, 3.60179907684385857083629250787, 4.54480057112590762336125664297, 6.18914890850467584494375401152, 6.88515186790874490901788660610, 7.76743997998954436023378580566, 9.269902351368633389064156209817, 10.41650051242350633299117182846

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