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Properties

Label 2-80-80.3-c1-0-6
Degree $2$
Conductor $80$
Sign $0.994 - 0.109i$
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  + (1.41 + 0.0660i)2-s − 0.496·3-s + (1.99 + 0.186i)4-s + (−2.00 + 0.987i)5-s + (−0.701 − 0.0328i)6-s + (1.55 − 1.55i)7-s + (2.80 + 0.395i)8-s − 2.75·9-s + (−2.89 + 1.26i)10-s + (−4.19 − 4.19i)11-s + (−0.988 − 0.0927i)12-s + 5.09i·13-s + (2.29 − 2.09i)14-s + (0.996 − 0.490i)15-s + (3.93 + 0.743i)16-s + (0.213 − 0.213i)17-s + ⋯
L(s)  = 1  + (0.998 + 0.0467i)2-s − 0.286·3-s + (0.995 + 0.0933i)4-s + (−0.897 + 0.441i)5-s + (−0.286 − 0.0133i)6-s + (0.587 − 0.587i)7-s + (0.990 + 0.139i)8-s − 0.917·9-s + (−0.916 + 0.399i)10-s + (−1.26 − 1.26i)11-s + (−0.285 − 0.0267i)12-s + 1.41i·13-s + (0.614 − 0.559i)14-s + (0.257 − 0.126i)15-s + (0.982 + 0.185i)16-s + (0.0517 − 0.0517i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.994 - 0.109i$
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: \((2,\ 80,\ (\ :1/2),\ 0.994 - 0.109i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33117 + 0.0728634i\)
\(L(\frac12)\) \(\approx\) \(1.33117 + 0.0728634i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0660i)T \)
5 \( 1 + (2.00 - 0.987i)T \)
good3 \( 1 + 0.496T + 3T^{2} \)
7 \( 1 + (-1.55 + 1.55i)T - 7iT^{2} \)
11 \( 1 + (4.19 + 4.19i)T + 11iT^{2} \)
13 \( 1 - 5.09iT - 13T^{2} \)
17 \( 1 + (-0.213 + 0.213i)T - 17iT^{2} \)
19 \( 1 + (-0.844 - 0.844i)T + 19iT^{2} \)
23 \( 1 + (-1.70 - 1.70i)T + 23iT^{2} \)
29 \( 1 + (-2.24 + 2.24i)T - 29iT^{2} \)
31 \( 1 + 0.818iT - 31T^{2} \)
37 \( 1 - 5.12iT - 37T^{2} \)
41 \( 1 + 3.34iT - 41T^{2} \)
43 \( 1 + 4.49iT - 43T^{2} \)
47 \( 1 + (-4.29 - 4.29i)T + 47iT^{2} \)
53 \( 1 + 1.00T + 53T^{2} \)
59 \( 1 + (7.65 - 7.65i)T - 59iT^{2} \)
61 \( 1 + (1.90 + 1.90i)T + 61iT^{2} \)
67 \( 1 + 11.0iT - 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + (-2.70 + 2.70i)T - 73iT^{2} \)
79 \( 1 + 8.32T + 79T^{2} \)
83 \( 1 + 9.17T + 83T^{2} \)
89 \( 1 + 4.25T + 89T^{2} \)
97 \( 1 + (7.15 - 7.15i)T - 97iT^{2} \)
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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

−14.15685738139674212737010779481, −13.69297980386297648504775100503, −12.07366657820142949147814534885, −11.26328412528775299033848077462, −10.71534314597435587853256524963, −8.357535148674643010251516203813, −7.32166178882836081414107368594, −5.93215966959188736103960503054, −4.54159301100008861327880128511, −3.06187346871391095084032398811, 2.84000252646329249803409146391, 4.79443285457722772839826610075, 5.53037620697632889151644035045, 7.42298903810641290860573678287, 8.358868483545534764774981994394, 10.42162312147607482937588967410, 11.39500172540144681204082650881, 12.37901667288690164834641086643, 12.98025049548092383137913306227, 14.59855240063187609005930318053

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