Properties

Label 2-80-80.43-c1-0-3
Degree $2$
Conductor $80$
Sign $0.104 - 0.994i$
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  + (0.687 + 1.23i)2-s + 0.614i·3-s + (−1.05 + 1.69i)4-s + (−2.07 + 0.832i)5-s + (−0.759 + 0.422i)6-s + (2.83 − 2.83i)7-s + (−2.82 − 0.134i)8-s + 2.62·9-s + (−2.45 − 1.99i)10-s + (1.95 − 1.95i)11-s + (−1.04 − 0.647i)12-s − 2.05·13-s + (5.45 + 1.55i)14-s + (−0.511 − 1.27i)15-s + (−1.77 − 3.58i)16-s + (−4.06 + 4.06i)17-s + ⋯
L(s)  = 1  + (0.486 + 0.873i)2-s + 0.354i·3-s + (−0.527 + 0.849i)4-s + (−0.928 + 0.372i)5-s + (−0.310 + 0.172i)6-s + (1.07 − 1.07i)7-s + (−0.998 − 0.0473i)8-s + 0.874·9-s + (−0.776 − 0.630i)10-s + (0.590 − 0.590i)11-s + (−0.301 − 0.187i)12-s − 0.569·13-s + (1.45 + 0.415i)14-s + (−0.132 − 0.329i)15-s + (−0.444 − 0.895i)16-s + (−0.986 + 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.104 - 0.994i$
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: \((2,\ 80,\ (\ :1/2),\ 0.104 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.812951 + 0.732058i\)
\(L(\frac12)\) \(\approx\) \(0.812951 + 0.732058i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.687 - 1.23i)T \)
5 \( 1 + (2.07 - 0.832i)T \)
good3 \( 1 - 0.614iT - 3T^{2} \)
7 \( 1 + (-2.83 + 2.83i)T - 7iT^{2} \)
11 \( 1 + (-1.95 + 1.95i)T - 11iT^{2} \)
13 \( 1 + 2.05T + 13T^{2} \)
17 \( 1 + (4.06 - 4.06i)T - 17iT^{2} \)
19 \( 1 + (-0.683 + 0.683i)T - 19iT^{2} \)
23 \( 1 + (4.95 + 4.95i)T + 23iT^{2} \)
29 \( 1 + (0.835 + 0.835i)T + 29iT^{2} \)
31 \( 1 - 2.35iT - 31T^{2} \)
37 \( 1 + 4.54T + 37T^{2} \)
41 \( 1 - 5.07iT - 41T^{2} \)
43 \( 1 + 0.849T + 43T^{2} \)
47 \( 1 + (-2.72 - 2.72i)T + 47iT^{2} \)
53 \( 1 - 5.17iT - 53T^{2} \)
59 \( 1 + (-4.16 - 4.16i)T + 59iT^{2} \)
61 \( 1 + (-5.55 + 5.55i)T - 61iT^{2} \)
67 \( 1 + 1.73T + 67T^{2} \)
71 \( 1 - 2.33T + 71T^{2} \)
73 \( 1 + (-4.39 + 4.39i)T - 73iT^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 2.75iT - 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + (3.52 - 3.52i)T - 97iT^{2} \)
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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

−14.68741333576407925443563310353, −13.95361815109619134226275436106, −12.64279417238456249228974427633, −11.44137856047894984991471362114, −10.37718459681003917039633564326, −8.585104660251373589008184281129, −7.61125761896255316115618066976, −6.63123374325369996593243073490, −4.58162580949885439806168679542, −3.93047030287500441640760317790, 1.95813183540523251651850835743, 4.17535766716212241244810693878, 5.22958028383356760367181170569, 7.21104873352215385582307598686, 8.630440152697391424305050682451, 9.755211554658424139736326603860, 11.43281578419205298351996257676, 11.91373869665796178434521418583, 12.73242113511172795928592907099, 14.03258096595873744640418610776

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