Properties

Label 2-1080-1.1-c1-0-6
Degree $2$
Conductor $1080$
Sign $1$
Analytic cond. $8.62384$
Root an. cond. $2.93663$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·11-s + 3·17-s − 19-s + 3·23-s + 25-s + 4·29-s − 5·31-s + 10·37-s + 6·41-s − 6·43-s + 8·47-s − 7·49-s + 3·53-s + 2·55-s + 5·61-s − 2·67-s + 2·71-s + 6·73-s − 11·79-s + 9·83-s + 3·85-s + 10·89-s − 95-s + 8·97-s − 12·101-s − 12·103-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.603·11-s + 0.727·17-s − 0.229·19-s + 0.625·23-s + 1/5·25-s + 0.742·29-s − 0.898·31-s + 1.64·37-s + 0.937·41-s − 0.914·43-s + 1.16·47-s − 49-s + 0.412·53-s + 0.269·55-s + 0.640·61-s − 0.244·67-s + 0.237·71-s + 0.702·73-s − 1.23·79-s + 0.987·83-s + 0.325·85-s + 1.05·89-s − 0.102·95-s + 0.812·97-s − 1.19·101-s − 1.18·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.865641483\)
\(L(\frac12)\) \(\approx\) \(1.865641483\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−9.773258117712398312086363543185, −9.177757084421316742601028107132, −8.273121818092208623524729034923, −7.35341216339073951945123749294, −6.46183002954720332042957563162, −5.67617773523253624389817414621, −4.68654110242436923100068599220, −3.62324019465866523110556351352, −2.47913934577997053460459890258, −1.13214773225246297442393721548, 1.13214773225246297442393721548, 2.47913934577997053460459890258, 3.62324019465866523110556351352, 4.68654110242436923100068599220, 5.67617773523253624389817414621, 6.46183002954720332042957563162, 7.35341216339073951945123749294, 8.273121818092208623524729034923, 9.177757084421316742601028107132, 9.773258117712398312086363543185

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