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Properties

Label 2-80-20.7-c1-0-0
Degree $2$
Conductor $80$
Sign $0.995 - 0.0898i$
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  + (2 + i)5-s − 3i·9-s + (−5 + 5i)13-s + (−5 − 5i)17-s + (3 + 4i)25-s − 4i·29-s + (5 + 5i)37-s + 8·41-s + (3 − 6i)45-s + 7i·49-s + (5 − 5i)53-s − 12·61-s + (−15 + 5i)65-s + (5 − 5i)73-s − 9·81-s + ⋯
L(s)  = 1  + (0.894 + 0.447i)5-s − i·9-s + (−1.38 + 1.38i)13-s + (−1.21 − 1.21i)17-s + (0.600 + 0.800i)25-s − 0.742i·29-s + (0.821 + 0.821i)37-s + 1.24·41-s + (0.447 − 0.894i)45-s + i·49-s + (0.686 − 0.686i)53-s − 1.53·61-s + (−1.86 + 0.620i)65-s + (0.585 − 0.585i)73-s − 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.995 - 0.0898i$
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: \((2,\ 80,\ (\ :1/2),\ 0.995 - 0.0898i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01291 + 0.0455749i\)
\(L(\frac12)\) \(\approx\) \(1.01291 + 0.0455749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - i)T \)
good3 \( 1 + 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (5 - 5i)T - 13iT^{2} \)
17 \( 1 + (5 + 5i)T + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-5 - 5i)T + 37iT^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 + 5i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 + (-5 - 5i)T + 97iT^{2} \)
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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

−14.36196059830176043760838360071, −13.52087354184667345390636775401, −12.18903620283370482013499177600, −11.22182095132037467626831895479, −9.674845215232649046593868163176, −9.215903905955930506680129574560, −7.18411973635644872290120913485, −6.29307059904154622423476335696, −4.59227006210364649411590411646, −2.49781743409028979957278186652, 2.33847888997287159193861359754, 4.75363631493516026702961808663, 5.87142516828485734671431238577, 7.53894588705916063426304420155, 8.755222109170443434349305797563, 10.06377174762825158964122584652, 10.85059261549145160635221462498, 12.60039612390521967293181515483, 13.13304650984422156221169451965, 14.32719116835131344129120718687

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