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Properties

Label 2-80-5.4-c19-0-3
Degree $2$
Conductor $80$
Sign $0.831 + 0.556i$
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  − 6.18e4i·3-s + (−3.63e6 − 2.42e6i)5-s − 1.58e8i·7-s − 2.66e9·9-s − 2.71e9·11-s + 3.10e10i·13-s + (−1.50e11 + 2.24e11i)15-s − 9.10e11i·17-s + 2.23e11·19-s − 9.83e12·21-s + 9.90e12i·23-s + (7.28e12 + 1.76e13i)25-s + 9.31e13i·27-s − 3.13e13·29-s + 7.66e13·31-s + ⋯
L(s)  = 1  − 1.81i·3-s + (−0.831 − 0.556i)5-s − 1.48i·7-s − 2.29·9-s − 0.347·11-s + 0.812i·13-s + (−1.00 + 1.50i)15-s − 1.86i·17-s + 0.158·19-s − 2.70·21-s + 1.14i·23-s + (0.381 + 0.924i)25-s + 2.34i·27-s − 0.401·29-s + 0.520·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.556i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.831 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.831 + 0.556i$
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: \((2,\ 80,\ (\ :19/2),\ 0.831 + 0.556i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.4386517871\)
\(L(\frac12)\) \(\approx\) \(0.4386517871\)
\(L(\frac{21}{2})\) 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 \( 1 + (3.63e6 + 2.42e6i)T \)
good3 \( 1 + 6.18e4iT - 1.16e9T^{2} \)
7 \( 1 + 1.58e8iT - 1.13e16T^{2} \)
11 \( 1 + 2.71e9T + 6.11e19T^{2} \)
13 \( 1 - 3.10e10iT - 1.46e21T^{2} \)
17 \( 1 + 9.10e11iT - 2.39e23T^{2} \)
19 \( 1 - 2.23e11T + 1.97e24T^{2} \)
23 \( 1 - 9.90e12iT - 7.46e25T^{2} \)
29 \( 1 + 3.13e13T + 6.10e27T^{2} \)
31 \( 1 - 7.66e13T + 2.16e28T^{2} \)
37 \( 1 + 2.68e14iT - 6.24e29T^{2} \)
41 \( 1 - 1.01e15T + 4.39e30T^{2} \)
43 \( 1 + 3.06e14iT - 1.08e31T^{2} \)
47 \( 1 - 5.12e15iT - 5.88e31T^{2} \)
53 \( 1 - 1.31e16iT - 5.77e32T^{2} \)
59 \( 1 - 9.88e16T + 4.42e33T^{2} \)
61 \( 1 - 2.91e16T + 8.34e33T^{2} \)
67 \( 1 + 8.44e16iT - 4.95e34T^{2} \)
71 \( 1 - 1.55e17T + 1.49e35T^{2} \)
73 \( 1 - 9.23e17iT - 2.53e35T^{2} \)
79 \( 1 + 9.07e17T + 1.13e36T^{2} \)
83 \( 1 - 2.02e18iT - 2.90e36T^{2} \)
89 \( 1 + 4.68e18T + 1.09e37T^{2} \)
97 \( 1 - 2.25e18iT - 5.60e37T^{2} \)
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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

−11.21568606612549359497687311308, −9.416187829499394164449432105751, −8.099526356975571736261839481149, −7.30996428478882134894430033737, −6.93467147861600107674622139255, −5.31631092469333212539394027780, −4.00858182045291331719475210152, −2.69719667789535844290836434643, −1.31064026399520435739991887579, −0.74731317648709150965300767353, 0.11469942033208262416040094694, 2.43398493526652403382901861713, 3.27731955236770016644318774627, 4.19762494303254053471176321709, 5.29339881175850043735234706921, 6.16895529428011169057727433989, 8.195845166845989781173632586631, 8.707191810466035707886825020653, 10.04729744909149402091038945996, 10.69947975418987704266690102219

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