Properties

Label 2-80-80.43-c1-0-8
Degree $2$
Conductor $80$
Sign $-0.297 + 0.954i$
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  + (−0.0660 − 1.41i)2-s − 0.496i·3-s + (−1.99 + 0.186i)4-s + (−0.987 − 2.00i)5-s + (−0.701 + 0.0328i)6-s + (1.55 − 1.55i)7-s + (0.395 + 2.80i)8-s + 2.75·9-s + (−2.76 + 1.52i)10-s + (−4.19 + 4.19i)11-s + (0.0927 + 0.988i)12-s + 5.09·13-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  + (−0.0467 − 0.998i)2-s − 0.286i·3-s + (−0.995 + 0.0933i)4-s + (−0.441 − 0.897i)5-s + (−0.286 + 0.0133i)6-s + (0.587 − 0.587i)7-s + (0.139 + 0.990i)8-s + 0.917·9-s + (−0.875 + 0.482i)10-s + (−1.26 + 1.26i)11-s + (0.0267 + 0.285i)12-s + 1.41·13-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.297 + 0.954i$
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: \((2,\ 80,\ (\ :1/2),\ -0.297 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.510684 - 0.693990i\)
\(L(\frac12)\) \(\approx\) \(0.510684 - 0.693990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−13.42949227417018672847499534794, −13.03462503226843705109011184835, −11.99534869686843027643777202302, −10.81899674984462984771507647415, −9.844593377480769660702391727111, −8.422872581917627559009677318301, −7.49765615840283016594913313476, −5.06364954192457090886325553587, −4.02379646555656430446560006042, −1.54532472257734829377951785688, 3.59452952640116843501973373663, 5.26148760688914238905177044081, 6.53539307115159497819393723687, 7.88343028593941408125589062355, 8.720407871688205963211510433731, 10.36108516246407036592889439226, 11.17907618318975396542052236527, 12.92884301188020011244122391704, 13.86141517107041202774782899851, 15.00254247836111967447988430152

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