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Properties

Label 2-80-80.43-c1-0-0
Degree $2$
Conductor $80$
Sign $-0.163 - 0.986i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.31 + 0.516i)2-s + 1.28i·3-s + (1.46 − 1.36i)4-s + (−0.841 + 2.07i)5-s + (−0.662 − 1.68i)6-s + (−1.13 + 1.13i)7-s + (−1.22 + 2.54i)8-s + 1.35·9-s + (0.0374 − 3.16i)10-s + (−2.32 + 2.32i)11-s + (1.74 + 1.87i)12-s + 1.36·13-s + (0.911 − 2.08i)14-s + (−2.65 − 1.07i)15-s + (0.297 − 3.98i)16-s + (5.25 − 5.25i)17-s + ⋯
L(s)  = 1  + (−0.930 + 0.365i)2-s + 0.739i·3-s + (0.732 − 0.680i)4-s + (−0.376 + 0.926i)5-s + (−0.270 − 0.688i)6-s + (−0.430 + 0.430i)7-s + (−0.433 + 0.901i)8-s + 0.452·9-s + (0.0118 − 0.999i)10-s + (−0.700 + 0.700i)11-s + (0.503 + 0.542i)12-s + 0.378·13-s + (0.243 − 0.558i)14-s + (−0.685 − 0.278i)15-s + (0.0744 − 0.997i)16-s + (1.27 − 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.163 - 0.986i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ -0.163 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.394719 + 0.465464i\)
\(L(\frac12)\) \(\approx\) \(0.394719 + 0.465464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.516i)T \)
5 \( 1 + (0.841 - 2.07i)T \)
good3 \( 1 - 1.28iT - 3T^{2} \)
7 \( 1 + (1.13 - 1.13i)T - 7iT^{2} \)
11 \( 1 + (2.32 - 2.32i)T - 11iT^{2} \)
13 \( 1 - 1.36T + 13T^{2} \)
17 \( 1 + (-5.25 + 5.25i)T - 17iT^{2} \)
19 \( 1 + (-3.69 + 3.69i)T - 19iT^{2} \)
23 \( 1 + (0.911 + 0.911i)T + 23iT^{2} \)
29 \( 1 + (-2.37 - 2.37i)T + 29iT^{2} \)
31 \( 1 - 0.242iT - 31T^{2} \)
37 \( 1 + 3.34T + 37T^{2} \)
41 \( 1 - 2.66iT - 41T^{2} \)
43 \( 1 - 9.04T + 43T^{2} \)
47 \( 1 + (7.87 + 7.87i)T + 47iT^{2} \)
53 \( 1 + 5.80iT - 53T^{2} \)
59 \( 1 + (5.91 + 5.91i)T + 59iT^{2} \)
61 \( 1 + (6.67 - 6.67i)T - 61iT^{2} \)
67 \( 1 + 4.54T + 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + (-1.49 + 1.49i)T - 73iT^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 3.26iT - 83T^{2} \)
89 \( 1 + 9.77T + 89T^{2} \)
97 \( 1 + (1.63 - 1.63i)T - 97iT^{2} \)
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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

−15.15932606070502318945431472621, −14.02805594323753062792018916731, −12.21428069462728995071142033821, −11.03668133936408248918926184080, −10.06025704089410942463344961604, −9.396677235921343977276814881120, −7.74538632520916891218735327153, −6.83531060610601045320961628960, −5.18435826076035459274526137002, −2.98832188762392502769356523350, 1.22425056110779022660606694733, 3.65101419125624917640561548549, 6.01915377922946291415264831732, 7.64376204382657550924917195632, 8.176256323995569430028925829027, 9.644481142185433228785616939600, 10.67835060902780995792567351507, 12.13207697509575099639504718796, 12.68189945282010254257699264928, 13.70791419922416758771387381333

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