Properties

Label 2-80-1.1-c5-0-7
Degree $2$
Conductor $80$
Sign $-1$
Analytic cond. $12.8307$
Root an. cond. $3.58199$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 25·5-s + 62·7-s − 239·9-s + 144·11-s − 654·13-s − 50·15-s − 1.19e3·17-s − 556·19-s + 124·21-s − 2.18e3·23-s + 625·25-s − 964·27-s − 1.57e3·29-s − 9.66e3·31-s + 288·33-s − 1.55e3·35-s − 3.53e3·37-s − 1.30e3·39-s + 7.46e3·41-s + 7.11e3·43-s + 5.97e3·45-s + 2.82e4·47-s − 1.29e4·49-s − 2.38e3·51-s − 1.30e4·53-s − 3.60e3·55-s + ⋯
L(s)  = 1  + 0.128·3-s − 0.447·5-s + 0.478·7-s − 0.983·9-s + 0.358·11-s − 1.07·13-s − 0.0573·15-s − 0.998·17-s − 0.353·19-s + 0.0613·21-s − 0.860·23-s + 1/5·25-s − 0.254·27-s − 0.348·29-s − 1.80·31-s + 0.0460·33-s − 0.213·35-s − 0.424·37-s − 0.137·39-s + 0.693·41-s + 0.586·43-s + 0.439·45-s + 1.86·47-s − 0.771·49-s − 0.128·51-s − 0.637·53-s − 0.160·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-1$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + p^{2} T \)
good3 \( 1 - 2 T + p^{5} T^{2} \)
7 \( 1 - 62 T + p^{5} T^{2} \)
11 \( 1 - 144 T + p^{5} T^{2} \)
13 \( 1 + 654 T + p^{5} T^{2} \)
17 \( 1 + 70 p T + p^{5} T^{2} \)
19 \( 1 + 556 T + p^{5} T^{2} \)
23 \( 1 + 2182 T + p^{5} T^{2} \)
29 \( 1 + 1578 T + p^{5} T^{2} \)
31 \( 1 + 9660 T + p^{5} T^{2} \)
37 \( 1 + 3534 T + p^{5} T^{2} \)
41 \( 1 - 182 p T + p^{5} T^{2} \)
43 \( 1 - 7114 T + p^{5} T^{2} \)
47 \( 1 - 602 p T + p^{5} T^{2} \)
53 \( 1 + 13046 T + p^{5} T^{2} \)
59 \( 1 - 37092 T + p^{5} T^{2} \)
61 \( 1 - 39570 T + p^{5} T^{2} \)
67 \( 1 - 56734 T + p^{5} T^{2} \)
71 \( 1 + 45588 T + p^{5} T^{2} \)
73 \( 1 - 11842 T + p^{5} T^{2} \)
79 \( 1 + 94216 T + p^{5} T^{2} \)
83 \( 1 - 31482 T + p^{5} T^{2} \)
89 \( 1 + 94054 T + p^{5} T^{2} \)
97 \( 1 - 23714 T + p^{5} T^{2} \)
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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

−12.77970551427644658406719118298, −11.70077686225628205683325412548, −10.86632866059897996605927953634, −9.329515278334569492283032204466, −8.296996535665789670340641076261, −7.10653125663884277045602798918, −5.52386276439639178888496163063, −4.06099955947417801885676913305, −2.29762831469545964195930172582, 0, 2.29762831469545964195930172582, 4.06099955947417801885676913305, 5.52386276439639178888496163063, 7.10653125663884277045602798918, 8.296996535665789670340641076261, 9.329515278334569492283032204466, 10.86632866059897996605927953634, 11.70077686225628205683325412548, 12.77970551427644658406719118298

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