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Properties

Label 2-80-20.7-c1-0-2
Degree $2$
Conductor $80$
Sign $-0.0299 + 0.999i$
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  + (−1.73 − 1.73i)3-s + (−1 − 2i)5-s + (1.73 − 1.73i)7-s + 2.99i·9-s + 3.46i·11-s + (1 − i)13-s + (−1.73 + 5.19i)15-s + (1 + i)17-s + 6.92·19-s − 5.99·21-s + (−1.73 − 1.73i)23-s + (−3 + 4i)25-s − 4i·29-s − 3.46i·31-s + (5.99 − 5.99i)33-s + ⋯
L(s)  = 1  + (−0.999 − 0.999i)3-s + (−0.447 − 0.894i)5-s + (0.654 − 0.654i)7-s + 0.999i·9-s + 1.04i·11-s + (0.277 − 0.277i)13-s + (−0.447 + 1.34i)15-s + (0.242 + 0.242i)17-s + 1.58·19-s − 1.30·21-s + (−0.361 − 0.361i)23-s + (−0.600 + 0.800i)25-s − 0.742i·29-s − 0.622i·31-s + (1.04 − 1.04i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.0299 + 0.999i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ -0.0299 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.491552 - 0.506512i\)
\(L(\frac12)\) \(\approx\) \(0.491552 - 0.506512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1 + 2i)T \)
good3 \( 1 + (1.73 + 1.73i)T + 3iT^{2} \)
7 \( 1 + (-1.73 + 1.73i)T - 7iT^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (-1 - i)T + 17iT^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 + (1.73 + 1.73i)T + 23iT^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (-5 - 5i)T + 37iT^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + (1.73 + 1.73i)T + 43iT^{2} \)
47 \( 1 + (-1.73 + 1.73i)T - 47iT^{2} \)
53 \( 1 + (7 - 7i)T - 53iT^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + (5.19 - 5.19i)T - 67iT^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (7 - 7i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-12.1 - 12.1i)T + 83iT^{2} \)
89 \( 1 + 8iT - 89T^{2} \)
97 \( 1 + (7 + 7i)T + 97iT^{2} \)
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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

−13.81633263773450965288797748318, −12.82061118200499914857382541786, −12.00769583087360798257596335880, −11.24871481432825643109822433450, −9.757738081556957364521266079889, −8.000128972104114014046112469308, −7.26250094854783193422925026063, −5.70724368024664258813509564284, −4.44803129710903846605872563753, −1.21171864887547730782268775920, 3.42667385456365767720831796605, 5.06669545254652433448097839828, 6.11690165593306375463584261633, 7.77689947267413755636464967549, 9.295272979974243628707431541317, 10.58256322170514485076474245011, 11.35617472361586123844384568107, 11.92056483427757098338139318191, 13.87945765887155880541331514453, 14.81327340717866753813452054973

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