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Properties

Label 2-80-5.4-c19-0-52
Degree $2$
Conductor $80$
Sign $-0.435 - 0.900i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 332. i·3-s + (−1.90e6 − 3.93e6i)5-s − 1.13e8i·7-s + 1.16e9·9-s − 1.45e10·11-s + 1.38e10i·13-s + (1.30e9 − 6.31e8i)15-s − 5.92e11i·17-s + 1.47e12·19-s + 3.78e10·21-s − 9.86e12i·23-s + (−1.18e13 + 1.49e13i)25-s + 7.71e11i·27-s − 9.50e13·29-s − 1.31e14·31-s + ⋯
L(s)  = 1  + 0.00973i·3-s + (−0.435 − 0.900i)5-s − 1.06i·7-s + 0.999·9-s − 1.86·11-s + 0.361i·13-s + (0.00876 − 0.00424i)15-s − 1.21i·17-s + 1.04·19-s + 0.0103·21-s − 1.14i·23-s + (−0.620 + 0.784i)25-s + 0.0194i·27-s − 1.21·29-s − 0.894·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.435 - 0.900i$
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: \((2,\ 80,\ (\ :19/2),\ -0.435 - 0.900i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.6135940155\)
\(L(\frac12)\) \(\approx\) \(0.6135940155\)
\(L(\frac{21}{2})\) 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 \( 1 + (1.90e6 + 3.93e6i)T \)
good3 \( 1 - 332. iT - 1.16e9T^{2} \)
7 \( 1 + 1.13e8iT - 1.13e16T^{2} \)
11 \( 1 + 1.45e10T + 6.11e19T^{2} \)
13 \( 1 - 1.38e10iT - 1.46e21T^{2} \)
17 \( 1 + 5.92e11iT - 2.39e23T^{2} \)
19 \( 1 - 1.47e12T + 1.97e24T^{2} \)
23 \( 1 + 9.86e12iT - 7.46e25T^{2} \)
29 \( 1 + 9.50e13T + 6.10e27T^{2} \)
31 \( 1 + 1.31e14T + 2.16e28T^{2} \)
37 \( 1 + 8.24e14iT - 6.24e29T^{2} \)
41 \( 1 - 3.65e13T + 4.39e30T^{2} \)
43 \( 1 + 4.59e15iT - 1.08e31T^{2} \)
47 \( 1 + 7.10e15iT - 5.88e31T^{2} \)
53 \( 1 - 1.67e16iT - 5.77e32T^{2} \)
59 \( 1 + 9.74e16T + 4.42e33T^{2} \)
61 \( 1 - 1.79e16T + 8.34e33T^{2} \)
67 \( 1 - 6.41e15iT - 4.95e34T^{2} \)
71 \( 1 - 3.56e17T + 1.49e35T^{2} \)
73 \( 1 + 5.93e17iT - 2.53e35T^{2} \)
79 \( 1 + 1.23e18T + 1.13e36T^{2} \)
83 \( 1 - 1.04e18iT - 2.90e36T^{2} \)
89 \( 1 - 5.26e18T + 1.09e37T^{2} \)
97 \( 1 + 5.71e18iT - 5.60e37T^{2} \)
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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

−10.11622509774579838687580214864, −9.053478380946349191816178688113, −7.55875222047502198620678280558, −7.33828898164124985722349880274, −5.36527770510484533655053199098, −4.61003010686349427541718447965, −3.57303374230600260292486395585, −2.07933071085596359833581552240, −0.76970575773013880450546602753, −0.14449634280252908082837136648, 1.60052178022893314534159685867, 2.68797950269027009948121296670, 3.58993106388491467228911730122, 5.09883245656047839910253624555, 6.01379225146776370455639311523, 7.45868684063345969639163279414, 7.994709353997830833810770293321, 9.564440211702237032502505069771, 10.47821967965278452278864302465, 11.39004408758276181802191834664

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