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Properties

Label 2-80-5.4-c19-0-55
Degree $2$
Conductor $80$
Sign $0.615 - 0.787i$
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  − 5.29e4i·3-s + (2.69e6 − 3.44e6i)5-s − 1.42e8i·7-s − 1.64e9·9-s − 9.89e9·11-s − 2.44e10i·13-s + (−1.82e11 − 1.42e11i)15-s + 1.66e11i·17-s − 4.11e11·19-s − 7.56e12·21-s − 1.04e13i·23-s + (−4.60e12 − 1.85e13i)25-s + 2.56e13i·27-s − 5.43e12·29-s − 1.78e14·31-s + ⋯
L(s)  = 1  − 1.55i·3-s + (0.615 − 0.787i)5-s − 1.33i·7-s − 1.41·9-s − 1.26·11-s − 0.640i·13-s + (−1.22 − 0.957i)15-s + 0.340i·17-s − 0.292·19-s − 2.07·21-s − 1.20i·23-s + (−0.241 − 0.970i)25-s + 0.646i·27-s − 0.0695·29-s − 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.615 - 0.787i$
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: \((2,\ 80,\ (\ :19/2),\ 0.615 - 0.787i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.9782542827\)
\(L(\frac12)\) \(\approx\) \(0.9782542827\)
\(L(\frac{21}{2})\) 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 \( 1 + (-2.69e6 + 3.44e6i)T \)
good3 \( 1 + 5.29e4iT - 1.16e9T^{2} \)
7 \( 1 + 1.42e8iT - 1.13e16T^{2} \)
11 \( 1 + 9.89e9T + 6.11e19T^{2} \)
13 \( 1 + 2.44e10iT - 1.46e21T^{2} \)
17 \( 1 - 1.66e11iT - 2.39e23T^{2} \)
19 \( 1 + 4.11e11T + 1.97e24T^{2} \)
23 \( 1 + 1.04e13iT - 7.46e25T^{2} \)
29 \( 1 + 5.43e12T + 6.10e27T^{2} \)
31 \( 1 + 1.78e14T + 2.16e28T^{2} \)
37 \( 1 - 1.55e15iT - 6.24e29T^{2} \)
41 \( 1 - 4.13e15T + 4.39e30T^{2} \)
43 \( 1 - 5.52e15iT - 1.08e31T^{2} \)
47 \( 1 + 1.14e16iT - 5.88e31T^{2} \)
53 \( 1 + 1.27e16iT - 5.77e32T^{2} \)
59 \( 1 - 2.18e16T + 4.42e33T^{2} \)
61 \( 1 + 3.03e16T + 8.34e33T^{2} \)
67 \( 1 + 1.36e17iT - 4.95e34T^{2} \)
71 \( 1 + 1.27e17T + 1.49e35T^{2} \)
73 \( 1 - 2.51e17iT - 2.53e35T^{2} \)
79 \( 1 - 3.11e17T + 1.13e36T^{2} \)
83 \( 1 + 2.08e18iT - 2.90e36T^{2} \)
89 \( 1 + 4.33e18T + 1.09e37T^{2} \)
97 \( 1 + 7.51e18iT - 5.60e37T^{2} \)
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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

−10.00204814430103400334461855385, −8.378169017826161598795628050918, −7.75383382662992060936970086944, −6.72883716256047101558274198736, −5.70125193003630862475663576698, −4.49581871830015537484650540645, −2.80968926836833303682665711569, −1.75211256646178634242093284152, −0.841992221088116137559422321473, −0.20332850311213777779050099137, 2.11509510694860009543611485039, 2.83209731899916261502481396221, 3.98110367496410743074978346674, 5.38031657633739836195730365290, 5.73917138440362224155456804206, 7.45104336132662769290913083726, 9.061934408222597064846500473157, 9.472379601414812694870363423251, 10.64266112623169825889624005856, 11.22870820330009839728770269152

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