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Properties

Label 2-80-80.3-c1-0-0
Degree $2$
Conductor $80$
Sign $-0.839 - 0.543i$
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  + (0.430 + 1.34i)2-s − 2.96·3-s + (−1.62 + 1.15i)4-s + (−0.177 + 2.22i)5-s + (−1.27 − 3.99i)6-s + (−0.115 + 0.115i)7-s + (−2.26 − 1.69i)8-s + 5.79·9-s + (−3.07 + 0.720i)10-s + (2.95 + 2.95i)11-s + (4.83 − 3.43i)12-s + 1.55i·13-s + (−0.204 − 0.105i)14-s + (0.525 − 6.61i)15-s + (1.31 − 3.77i)16-s + (0.299 − 0.299i)17-s + ⋯
L(s)  = 1  + (0.304 + 0.952i)2-s − 1.71·3-s + (−0.814 + 0.579i)4-s + (−0.0793 + 0.996i)5-s + (−0.520 − 1.63i)6-s + (−0.0435 + 0.0435i)7-s + (−0.800 − 0.599i)8-s + 1.93·9-s + (−0.973 + 0.227i)10-s + (0.892 + 0.892i)11-s + (1.39 − 0.992i)12-s + 0.432i·13-s + (−0.0546 − 0.0282i)14-s + (0.135 − 1.70i)15-s + (0.327 − 0.944i)16-s + (0.0726 − 0.0726i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.839 - 0.543i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ -0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.164922 + 0.558417i\)
\(L(\frac12)\) \(\approx\) \(0.164922 + 0.558417i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.430 - 1.34i)T \)
5 \( 1 + (0.177 - 2.22i)T \)
good3 \( 1 + 2.96T + 3T^{2} \)
7 \( 1 + (0.115 - 0.115i)T - 7iT^{2} \)
11 \( 1 + (-2.95 - 2.95i)T + 11iT^{2} \)
13 \( 1 - 1.55iT - 13T^{2} \)
17 \( 1 + (-0.299 + 0.299i)T - 17iT^{2} \)
19 \( 1 + (2.26 + 2.26i)T + 19iT^{2} \)
23 \( 1 + (-4.14 - 4.14i)T + 23iT^{2} \)
29 \( 1 + (-0.289 + 0.289i)T - 29iT^{2} \)
31 \( 1 - 4.18iT - 31T^{2} \)
37 \( 1 + 1.63iT - 37T^{2} \)
41 \( 1 + 7.61iT - 41T^{2} \)
43 \( 1 + 6.72iT - 43T^{2} \)
47 \( 1 + (-4.38 - 4.38i)T + 47iT^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + (-1.63 + 1.63i)T - 59iT^{2} \)
61 \( 1 + (1.23 + 1.23i)T + 61iT^{2} \)
67 \( 1 + 2.49iT - 67T^{2} \)
71 \( 1 - 8.00T + 71T^{2} \)
73 \( 1 + (1.12 - 1.12i)T - 73iT^{2} \)
79 \( 1 - 3.62T + 79T^{2} \)
83 \( 1 - 1.62T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (-9.69 + 9.69i)T - 97iT^{2} \)
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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

−15.08522907344100390543956213143, −13.91549983801695502107008539384, −12.52816476296601895860674385827, −11.72737978853206026884777954360, −10.62682574478976711029806455803, −9.328088808244823761777077118468, −7.17016833798032541184106816505, −6.70117402993890786866802353201, −5.49639045490722261691543526730, −4.15601253177353902389854066932, 0.916915229086749418346729581447, 4.14657684899582333219578443174, 5.30887662968168042618607220184, 6.28277968221160283579571724451, 8.566928556849731653148284917813, 9.911148282393493865064393454895, 11.00952335081074125359408640808, 11.77549741483412631279254026737, 12.56980963206724800310217852314, 13.39760190120175615566754698764

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