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Properties

Label 2-80-5.4-c19-0-37
Degree $2$
Conductor $80$
Sign $-0.508 + 0.860i$
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  − 3.31e4i·3-s + (2.22e6 − 3.75e6i)5-s − 2.70e7i·7-s + 6.28e7·9-s − 5.56e9·11-s + 4.21e10i·13-s + (−1.24e11 − 7.36e10i)15-s + 5.95e11i·17-s + 1.45e12·19-s − 8.95e11·21-s + 9.69e12i·23-s + (−9.19e12 − 1.67e13i)25-s − 4.06e13i·27-s + 9.46e13·29-s − 6.27e12·31-s + ⋯
L(s)  = 1  − 0.972i·3-s + (0.508 − 0.860i)5-s − 0.253i·7-s + 0.0540·9-s − 0.711·11-s + 1.10i·13-s + (−0.837 − 0.494i)15-s + 1.21i·17-s + 1.03·19-s − 0.246·21-s + 1.12i·23-s + (−0.482 − 0.876i)25-s − 1.02i·27-s + 1.21·29-s − 0.0426·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 + 0.860i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.508 + 0.860i$
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: \((2,\ 80,\ (\ :19/2),\ -0.508 + 0.860i)\)

Particular Values

\(L(10)\) \(\approx\) \(2.470814408\)
\(L(\frac12)\) \(\approx\) \(2.470814408\)
\(L(\frac{21}{2})\) 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 \( 1 + (-2.22e6 + 3.75e6i)T \)
good3 \( 1 + 3.31e4iT - 1.16e9T^{2} \)
7 \( 1 + 2.70e7iT - 1.13e16T^{2} \)
11 \( 1 + 5.56e9T + 6.11e19T^{2} \)
13 \( 1 - 4.21e10iT - 1.46e21T^{2} \)
17 \( 1 - 5.95e11iT - 2.39e23T^{2} \)
19 \( 1 - 1.45e12T + 1.97e24T^{2} \)
23 \( 1 - 9.69e12iT - 7.46e25T^{2} \)
29 \( 1 - 9.46e13T + 6.10e27T^{2} \)
31 \( 1 + 6.27e12T + 2.16e28T^{2} \)
37 \( 1 + 1.41e15iT - 6.24e29T^{2} \)
41 \( 1 + 9.33e14T + 4.39e30T^{2} \)
43 \( 1 - 2.15e13iT - 1.08e31T^{2} \)
47 \( 1 + 4.64e15iT - 5.88e31T^{2} \)
53 \( 1 + 3.42e16iT - 5.77e32T^{2} \)
59 \( 1 - 3.91e16T + 4.42e33T^{2} \)
61 \( 1 + 1.44e17T + 8.34e33T^{2} \)
67 \( 1 - 3.79e16iT - 4.95e34T^{2} \)
71 \( 1 - 7.32e17T + 1.49e35T^{2} \)
73 \( 1 + 5.16e17iT - 2.53e35T^{2} \)
79 \( 1 - 5.46e16T + 1.13e36T^{2} \)
83 \( 1 + 5.15e17iT - 2.90e36T^{2} \)
89 \( 1 - 3.08e18T + 1.09e37T^{2} \)
97 \( 1 + 5.49e18iT - 5.60e37T^{2} \)
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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

−10.35088540937994908435553418764, −9.301813545786286741509186360885, −8.174727509093752608280211719157, −7.23967458064053547875635266198, −6.15014355101715446175894152194, −5.07355547034290773715536218101, −3.84113974629238138505059998243, −2.15218014417988024754548862830, −1.47607535087204288036194036924, −0.52773454511329195048526008298, 0.933578837614708759124344403848, 2.64252102452058969966733552272, 3.16461104597031768895225727994, 4.69982710678500984207089488424, 5.49773871502739167545450483263, 6.78934020913959566569612142168, 7.940071139945141648293880089404, 9.354956146261804559228105963166, 10.18947325047233098273646031692, 10.74234334929472253664423698224

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