Properties

Label 2-540-1.1-c1-0-1
Degree $2$
Conductor $540$
Sign $1$
Analytic cond. $4.31192$
Root an. cond. $2.07651$
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 − 7-s + 6·11-s − 13-s − 19-s + 6·23-s + 25-s + 6·29-s + 8·31-s − 35-s − 7·37-s − 6·41-s − 4·43-s + 12·47-s − 6·49-s − 6·53-s + 6·55-s + 11·61-s − 65-s − 7·67-s − 6·71-s + 11·73-s − 6·77-s − 79-s + 6·83-s − 12·89-s + 91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 1.80·11-s − 0.277·13-s − 0.229·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.169·35-s − 1.15·37-s − 0.937·41-s − 0.609·43-s + 1.75·47-s − 6/7·49-s − 0.824·53-s + 0.809·55-s + 1.40·61-s − 0.124·65-s − 0.855·67-s − 0.712·71-s + 1.28·73-s − 0.683·77-s − 0.112·79-s + 0.658·83-s − 1.27·89-s + 0.104·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(4.31192\)
Root analytic conductor: \(2.07651\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.638542216\)
\(L(\frac12)\) \(\approx\) \(1.638542216\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−10.74794183187189613430165450110, −9.844564467715010164242704464375, −9.127418035079580641451740646574, −8.343543745788865357315277029134, −6.81615637958252312767893811403, −6.53488198804866315995421780383, −5.20522128847215086701609074104, −4.09617400536347052792491503589, −2.89403046630752699619618864147, −1.31220016809550134539290495913, 1.31220016809550134539290495913, 2.89403046630752699619618864147, 4.09617400536347052792491503589, 5.20522128847215086701609074104, 6.53488198804866315995421780383, 6.81615637958252312767893811403, 8.343543745788865357315277029134, 9.127418035079580641451740646574, 9.844564467715010164242704464375, 10.74794183187189613430165450110

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