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Properties

Label 2-80-5.4-c19-0-24
Degree $2$
Conductor $80$
Sign $0.743 - 0.668i$
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  + 1.93e4i·3-s + (−3.24e6 + 2.92e6i)5-s − 3.24e7i·7-s + 7.88e8·9-s − 2.83e9·11-s − 4.07e8i·13-s + (−5.64e10 − 6.27e10i)15-s + 1.61e11i·17-s − 1.92e11·19-s + 6.26e11·21-s + 5.34e12i·23-s + (2.00e12 − 1.89e13i)25-s + 3.77e13i·27-s + 1.07e14·29-s − 2.17e14·31-s + ⋯
L(s)  = 1  + 0.566i·3-s + (−0.743 + 0.668i)5-s − 0.303i·7-s + 0.678·9-s − 0.363·11-s − 0.0106i·13-s + (−0.379 − 0.421i)15-s + 0.331i·17-s − 0.136·19-s + 0.172·21-s + 0.618i·23-s + (0.105 − 0.994i)25-s + 0.951i·27-s + 1.37·29-s − 1.47·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.743 - 0.668i$
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: \((2,\ 80,\ (\ :19/2),\ 0.743 - 0.668i)\)

Particular Values

\(L(10)\) \(\approx\) \(1.751951244\)
\(L(\frac12)\) \(\approx\) \(1.751951244\)
\(L(\frac{21}{2})\) 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 \( 1 + (3.24e6 - 2.92e6i)T \)
good3 \( 1 - 1.93e4iT - 1.16e9T^{2} \)
7 \( 1 + 3.24e7iT - 1.13e16T^{2} \)
11 \( 1 + 2.83e9T + 6.11e19T^{2} \)
13 \( 1 + 4.07e8iT - 1.46e21T^{2} \)
17 \( 1 - 1.61e11iT - 2.39e23T^{2} \)
19 \( 1 + 1.92e11T + 1.97e24T^{2} \)
23 \( 1 - 5.34e12iT - 7.46e25T^{2} \)
29 \( 1 - 1.07e14T + 6.10e27T^{2} \)
31 \( 1 + 2.17e14T + 2.16e28T^{2} \)
37 \( 1 + 5.40e14iT - 6.24e29T^{2} \)
41 \( 1 + 9.49e14T + 4.39e30T^{2} \)
43 \( 1 + 5.49e15iT - 1.08e31T^{2} \)
47 \( 1 + 9.42e15iT - 5.88e31T^{2} \)
53 \( 1 + 9.09e15iT - 5.77e32T^{2} \)
59 \( 1 + 5.69e16T + 4.42e33T^{2} \)
61 \( 1 - 1.67e17T + 8.34e33T^{2} \)
67 \( 1 + 3.75e16iT - 4.95e34T^{2} \)
71 \( 1 + 3.92e17T + 1.49e35T^{2} \)
73 \( 1 - 3.40e17iT - 2.53e35T^{2} \)
79 \( 1 - 1.77e18T + 1.13e36T^{2} \)
83 \( 1 + 1.33e18iT - 2.90e36T^{2} \)
89 \( 1 - 1.37e18T + 1.09e37T^{2} \)
97 \( 1 + 1.13e19iT - 5.60e37T^{2} \)
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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

−10.68475356753262816184386594276, −10.16067442832930981787507020309, −8.797682500494161391929125608046, −7.57198066111644413563077720967, −6.80409414400115833232383113721, −5.29848050594285149336201836659, −4.09879953762100311651155903575, −3.43524600254193790574006128066, −2.03918301936979792190179269981, −0.56211096434190760185850472808, 0.59279030991058287194811851364, 1.49684880359779565505953955525, 2.78235311214066592954087969566, 4.16246192791726957242618959513, 5.07172593379825802827387274885, 6.47530661106941044018699756623, 7.54330000627091703260505324895, 8.358644908435680210098888756478, 9.481757499577552973414472283159, 10.77351117574257246736401913913

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