Properties

Label 2-81120-1.1-c1-0-41
Degree $2$
Conductor $81120$
Sign $-1$
Analytic cond. $647.746$
Root an. cond. $25.4508$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s + 4·11-s − 15-s − 2·17-s − 4·19-s + 8·23-s + 25-s − 27-s − 6·29-s − 4·33-s + 10·37-s − 6·41-s − 4·43-s + 45-s − 7·49-s + 2·51-s + 6·53-s + 4·55-s + 4·57-s − 12·59-s − 10·61-s + 12·67-s − 8·69-s + 12·71-s + 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.258·15-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.696·33-s + 1.64·37-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 49-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s − 1.56·59-s − 1.28·61-s + 1.46·67-s − 0.963·69-s + 1.42·71-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81120\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(647.746\)
Root analytic conductor: \(25.4508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 81120,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 + p T^{2} \) 1.7.a
11 \( 1 - 4 T + p T^{2} \) 1.11.ae
17 \( 1 + 2 T + p T^{2} \) 1.17.c
19 \( 1 + 4 T + p T^{2} \) 1.19.e
23 \( 1 - 8 T + p T^{2} \) 1.23.ai
29 \( 1 + 6 T + p T^{2} \) 1.29.g
31 \( 1 + p T^{2} \) 1.31.a
37 \( 1 - 10 T + p T^{2} \) 1.37.ak
41 \( 1 + 6 T + p T^{2} \) 1.41.g
43 \( 1 + 4 T + p T^{2} \) 1.43.e
47 \( 1 + p T^{2} \) 1.47.a
53 \( 1 - 6 T + p T^{2} \) 1.53.ag
59 \( 1 + 12 T + p T^{2} \) 1.59.m
61 \( 1 + 10 T + p T^{2} \) 1.61.k
67 \( 1 - 12 T + p T^{2} \) 1.67.am
71 \( 1 - 12 T + p T^{2} \) 1.71.am
73 \( 1 - 10 T + p T^{2} \) 1.73.ak
79 \( 1 - 8 T + p T^{2} \) 1.79.ai
83 \( 1 + 16 T + p T^{2} \) 1.83.q
89 \( 1 + 14 T + p T^{2} \) 1.89.o
97 \( 1 - 2 T + p T^{2} \) 1.97.ac
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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

−14.15775140663806, −13.77381936397007, −12.98447537265224, −12.88412638546304, −12.32763528877056, −11.57956946174260, −11.17015009609513, −10.99988763569419, −10.22213851195401, −9.682006304108123, −9.206014874739976, −8.863921855542933, −8.174428951739439, −7.523888355257506, −6.860529783633451, −6.461424907723106, −6.174328198630980, −5.325712465281563, −4.926486143022130, −4.276499054948290, −3.744331747213941, −3.024909791990722, −2.236196313338033, −1.570968174312003, −0.9689476328862510, 0, 0.9689476328862510, 1.570968174312003, 2.236196313338033, 3.024909791990722, 3.744331747213941, 4.276499054948290, 4.926486143022130, 5.325712465281563, 6.174328198630980, 6.461424907723106, 6.860529783633451, 7.523888355257506, 8.174428951739439, 8.863921855542933, 9.206014874739976, 9.682006304108123, 10.22213851195401, 10.99988763569419, 11.17015009609513, 11.57956946174260, 12.32763528877056, 12.88412638546304, 12.98447537265224, 13.77381936397007, 14.15775140663806

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