Properties

Label 2-80-80.43-c1-0-6
Degree $2$
Conductor $80$
Sign $0.987 - 0.160i$
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 + i)2-s − 2i·3-s + 2i·4-s + (1 − 2i)5-s + (2 − 2i)6-s + (−3 + 3i)7-s + (−2 + 2i)8-s − 9-s + (3 − i)10-s + (−1 + i)11-s + 4·12-s − 2·13-s − 6·14-s + (−4 − 2i)15-s − 4·16-s + (1 − i)17-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s − 1.15i·3-s + i·4-s + (0.447 − 0.894i)5-s + (0.816 − 0.816i)6-s + (−1.13 + 1.13i)7-s + (−0.707 + 0.707i)8-s − 0.333·9-s + (0.948 − 0.316i)10-s + (−0.301 + 0.301i)11-s + 1.15·12-s − 0.554·13-s − 1.60·14-s + (−1.03 − 0.516i)15-s − 16-s + (0.242 − 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.987 - 0.160i$
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: \((2,\ 80,\ (\ :1/2),\ 0.987 - 0.160i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25124 + 0.100864i\)
\(L(\frac12)\) \(\approx\) \(1.25124 + 0.100864i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
5 \( 1 + (-1 + 2i)T \)
good3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 + (1 - i)T - 11iT^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 + (-3 + 3i)T - 19iT^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 + (-7 - 7i)T + 29iT^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-7 - 7i)T + 47iT^{2} \)
53 \( 1 + 8iT - 53T^{2} \)
59 \( 1 + (3 + 3i)T + 59iT^{2} \)
61 \( 1 + (1 - i)T - 61iT^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3 - 3i)T - 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 2iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (11 - 11i)T - 97iT^{2} \)
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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

−14.15112524404569176674908270993, −13.18539645043385362545486359499, −12.49276997095376938063047430271, −12.06861790428375561600779554210, −9.592938614681951837097941131581, −8.524464693192186877471115792571, −7.19877950386906355931543697614, −6.16330575759644388198874314227, −5.04034024022624568110967431733, −2.64535399108189045264510170835, 3.10705146482163524261049807370, 4.09778591592978932596819751838, 5.73022578469574864023764171080, 7.06035091297824136120629226442, 9.563440182256676483416231456398, 10.22068358543928838428592717954, 10.66720740706958347951025042831, 12.18801456944399837843399013207, 13.56263174322409099297897186637, 14.13143550792508788546022118053

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