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Properties

Label 2-80-5.4-c19-0-43
Degree $2$
Conductor $80$
Sign $0.209 + 0.977i$
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  − 1.50e4i·3-s + (9.13e5 + 4.27e6i)5-s + 1.11e8i·7-s + 9.34e8·9-s + 5.13e9·11-s − 2.56e10i·13-s + (6.44e10 − 1.37e10i)15-s − 2.13e11i·17-s + 1.45e11·19-s + 1.68e12·21-s − 1.33e13i·23-s + (−1.74e13 + 7.80e12i)25-s − 3.16e13i·27-s − 6.12e13·29-s − 5.73e13·31-s + ⋯
L(s)  = 1  − 0.442i·3-s + (0.209 + 0.977i)5-s + 1.04i·7-s + 0.803·9-s + 0.656·11-s − 0.671i·13-s + (0.432 − 0.0926i)15-s − 0.437i·17-s + 0.103·19-s + 0.461·21-s − 1.54i·23-s + (−0.912 + 0.409i)25-s − 0.798i·27-s − 0.784·29-s − 0.389·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.209 + 0.977i$
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: \((2,\ 80,\ (\ :19/2),\ 0.209 + 0.977i)\)

Particular Values

\(L(10)\) \(\approx\) \(1.868149234\)
\(L(\frac12)\) \(\approx\) \(1.868149234\)
\(L(\frac{21}{2})\) 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 \( 1 + (-9.13e5 - 4.27e6i)T \)
good3 \( 1 + 1.50e4iT - 1.16e9T^{2} \)
7 \( 1 - 1.11e8iT - 1.13e16T^{2} \)
11 \( 1 - 5.13e9T + 6.11e19T^{2} \)
13 \( 1 + 2.56e10iT - 1.46e21T^{2} \)
17 \( 1 + 2.13e11iT - 2.39e23T^{2} \)
19 \( 1 - 1.45e11T + 1.97e24T^{2} \)
23 \( 1 + 1.33e13iT - 7.46e25T^{2} \)
29 \( 1 + 6.12e13T + 6.10e27T^{2} \)
31 \( 1 + 5.73e13T + 2.16e28T^{2} \)
37 \( 1 + 1.37e14iT - 6.24e29T^{2} \)
41 \( 1 + 2.75e15T + 4.39e30T^{2} \)
43 \( 1 - 1.86e14iT - 1.08e31T^{2} \)
47 \( 1 + 1.12e16iT - 5.88e31T^{2} \)
53 \( 1 + 1.94e16iT - 5.77e32T^{2} \)
59 \( 1 - 2.98e16T + 4.42e33T^{2} \)
61 \( 1 - 2.43e16T + 8.34e33T^{2} \)
67 \( 1 + 1.00e17iT - 4.95e34T^{2} \)
71 \( 1 - 1.53e16T + 1.49e35T^{2} \)
73 \( 1 - 4.79e16iT - 2.53e35T^{2} \)
79 \( 1 - 2.02e18T + 1.13e36T^{2} \)
83 \( 1 + 8.87e17iT - 2.90e36T^{2} \)
89 \( 1 + 3.09e18T + 1.09e37T^{2} \)
97 \( 1 + 1.08e19iT - 5.60e37T^{2} \)
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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

−10.53555417347413914937778265736, −9.572075210091360222336332413183, −8.372159685321539886167948439275, −7.13338784123609104092326040851, −6.39545614768384598002529379105, −5.26389821210441132505189664626, −3.74577767364950813831838287364, −2.57542334699286992867921730559, −1.76169520208872942367466297748, −0.34806560642638350467039654175, 1.10229021949538096147727671968, 1.66963144990051197235903898962, 3.71885091098777004869231499428, 4.26288930400932404215168512038, 5.37326531127781197538790100611, 6.77165930952890204930788032784, 7.78981286527087150779383433950, 9.177251246116349382370360104414, 9.773927492742627926783802920935, 10.92639085764065836598236369746

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