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Properties

Label 2-80-5.4-c19-0-46
Degree $2$
Conductor $80$
Sign $-0.496 + 0.867i$
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  − 1.41e4i·3-s + (−2.17e6 + 3.78e6i)5-s − 1.89e8i·7-s + 9.61e8·9-s + 6.57e9·11-s − 5.98e10i·13-s + (5.36e10 + 3.07e10i)15-s − 8.66e11i·17-s + 1.80e12·19-s − 2.68e12·21-s − 2.01e12i·23-s + (−9.65e12 − 1.64e13i)25-s − 3.00e13i·27-s + 7.93e13·29-s + 3.23e13·31-s + ⋯
L(s)  = 1  − 0.415i·3-s + (−0.496 + 0.867i)5-s − 1.77i·7-s + 0.827·9-s + 0.841·11-s − 1.56i·13-s + (0.360 + 0.206i)15-s − 1.77i·17-s + 1.28·19-s − 0.738·21-s − 0.233i·23-s + (−0.506 − 0.862i)25-s − 0.758i·27-s + 1.01·29-s + 0.219·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.867i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.496 + 0.867i$
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: \((2,\ 80,\ (\ :19/2),\ -0.496 + 0.867i)\)

Particular Values

\(L(10)\) \(\approx\) \(2.746155940\)
\(L(\frac12)\) \(\approx\) \(2.746155940\)
\(L(\frac{21}{2})\) 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 \( 1 + (2.17e6 - 3.78e6i)T \)
good3 \( 1 + 1.41e4iT - 1.16e9T^{2} \)
7 \( 1 + 1.89e8iT - 1.13e16T^{2} \)
11 \( 1 - 6.57e9T + 6.11e19T^{2} \)
13 \( 1 + 5.98e10iT - 1.46e21T^{2} \)
17 \( 1 + 8.66e11iT - 2.39e23T^{2} \)
19 \( 1 - 1.80e12T + 1.97e24T^{2} \)
23 \( 1 + 2.01e12iT - 7.46e25T^{2} \)
29 \( 1 - 7.93e13T + 6.10e27T^{2} \)
31 \( 1 - 3.23e13T + 2.16e28T^{2} \)
37 \( 1 + 6.92e13iT - 6.24e29T^{2} \)
41 \( 1 - 2.70e15T + 4.39e30T^{2} \)
43 \( 1 + 1.55e15iT - 1.08e31T^{2} \)
47 \( 1 - 1.13e16iT - 5.88e31T^{2} \)
53 \( 1 - 1.55e16iT - 5.77e32T^{2} \)
59 \( 1 - 1.03e16T + 4.42e33T^{2} \)
61 \( 1 - 7.24e16T + 8.34e33T^{2} \)
67 \( 1 + 3.74e16iT - 4.95e34T^{2} \)
71 \( 1 - 3.34e17T + 1.49e35T^{2} \)
73 \( 1 + 6.81e17iT - 2.53e35T^{2} \)
79 \( 1 - 1.19e18T + 1.13e36T^{2} \)
83 \( 1 - 5.37e17iT - 2.90e36T^{2} \)
89 \( 1 + 3.93e18T + 1.09e37T^{2} \)
97 \( 1 + 8.66e17iT - 5.60e37T^{2} \)
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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

−10.44501523494529937546179661709, −9.683067112601844454434736658941, −7.68214607278077473365166214332, −7.39967097870644341794973247365, −6.47702926048491419548095954852, −4.74013530749904331694775946498, −3.70287060107626412568090388259, −2.80082799944636999318959192983, −0.918365841589600403770233206421, −0.70763928547094721956273841000, 1.20507414442137182301804111880, 2.00373407972976731539097920295, 3.65561782311126310039081301383, 4.50162441347866769589248292789, 5.57334519451409318466508602053, 6.71663661611332197750076712450, 8.267221795758434195585392263250, 9.052881160285280146605947156853, 9.739542520008685829012460652335, 11.47206160467930543891351000484

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