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Properties

Label 2-80-5.4-c19-0-48
Degree $2$
Conductor $80$
Sign $-0.377 - 0.925i$
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  − 3.50e4i·3-s + (−1.65e6 − 4.04e6i)5-s + 1.45e8i·7-s − 6.73e7·9-s − 1.19e9·11-s − 1.78e10i·13-s + (−1.41e11 + 5.78e10i)15-s − 7.81e11i·17-s + 1.54e11·19-s + 5.10e12·21-s + 9.04e12i·23-s + (−1.36e13 + 1.33e13i)25-s − 3.83e13i·27-s + 6.31e13·29-s + 7.17e13·31-s + ⋯
L(s)  = 1  − 1.02i·3-s + (−0.377 − 0.925i)5-s + 1.36i·7-s − 0.0579·9-s − 0.152·11-s − 0.468i·13-s + (−0.952 + 0.388i)15-s − 1.59i·17-s + 0.109·19-s + 1.40·21-s + 1.04i·23-s + (−0.714 + 0.699i)25-s − 0.968i·27-s + 0.807·29-s + 0.487·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.377 - 0.925i$
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: \((2,\ 80,\ (\ :19/2),\ -0.377 - 0.925i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.2493255107\)
\(L(\frac12)\) \(\approx\) \(0.2493255107\)
\(L(\frac{21}{2})\) 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 \( 1 + (1.65e6 + 4.04e6i)T \)
good3 \( 1 + 3.50e4iT - 1.16e9T^{2} \)
7 \( 1 - 1.45e8iT - 1.13e16T^{2} \)
11 \( 1 + 1.19e9T + 6.11e19T^{2} \)
13 \( 1 + 1.78e10iT - 1.46e21T^{2} \)
17 \( 1 + 7.81e11iT - 2.39e23T^{2} \)
19 \( 1 - 1.54e11T + 1.97e24T^{2} \)
23 \( 1 - 9.04e12iT - 7.46e25T^{2} \)
29 \( 1 - 6.31e13T + 6.10e27T^{2} \)
31 \( 1 - 7.17e13T + 2.16e28T^{2} \)
37 \( 1 - 8.92e14iT - 6.24e29T^{2} \)
41 \( 1 + 2.19e15T + 4.39e30T^{2} \)
43 \( 1 + 6.25e15iT - 1.08e31T^{2} \)
47 \( 1 - 9.45e14iT - 5.88e31T^{2} \)
53 \( 1 + 4.13e16iT - 5.77e32T^{2} \)
59 \( 1 + 3.91e16T + 4.42e33T^{2} \)
61 \( 1 + 1.04e17T + 8.34e33T^{2} \)
67 \( 1 - 6.48e16iT - 4.95e34T^{2} \)
71 \( 1 - 2.44e17T + 1.49e35T^{2} \)
73 \( 1 + 9.90e16iT - 2.53e35T^{2} \)
79 \( 1 + 1.54e18T + 1.13e36T^{2} \)
83 \( 1 + 1.26e18iT - 2.90e36T^{2} \)
89 \( 1 + 3.89e18T + 1.09e37T^{2} \)
97 \( 1 + 8.43e18iT - 5.60e37T^{2} \)
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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

−9.834419750317448707993874561415, −8.784576960100678551702547294988, −7.956176317710245524535938313908, −6.92245962094330104324270715318, −5.61368905288929524792109400507, −4.82678709245391275800837839282, −3.11601221272162493213973625124, −2.02580742605401716440828149122, −1.02588203072733282338978633974, −0.05032728358381482530674401938, 1.38540267985156964643986252593, 2.96934183269010088387288541483, 4.02639464996911267741892538097, 4.45893548505471541409381778765, 6.23784145467900510823206175611, 7.18649923896175241600272046751, 8.279437154784458355565138096734, 9.774418896414055870345153180309, 10.56873564359091295843712896822, 10.95676765046604337033240440184

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