Properties

Label 2-271440-1.1-c1-0-16
Degree $2$
Conductor $271440$
Sign $1$
Analytic cond. $2167.45$
Root an. cond. $46.5559$
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 + 4·7-s − 4·11-s + 13-s + 6·17-s + 4·19-s − 8·23-s + 25-s − 29-s − 8·31-s − 4·35-s − 6·37-s + 2·41-s + 4·43-s + 9·49-s + 10·53-s + 4·55-s − 8·59-s − 10·61-s − 65-s + 8·67-s − 4·71-s + 14·73-s − 16·77-s + 16·79-s − 16·83-s − 6·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s − 1.20·11-s + 0.277·13-s + 1.45·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.185·29-s − 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.312·41-s + 0.609·43-s + 9/7·49-s + 1.37·53-s + 0.539·55-s − 1.04·59-s − 1.28·61-s − 0.124·65-s + 0.977·67-s − 0.474·71-s + 1.63·73-s − 1.82·77-s + 1.80·79-s − 1.75·83-s − 0.650·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 271440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 271440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(271440\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \cdot 29\)
Sign: $1$
Analytic conductor: \(2167.45\)
Root analytic conductor: \(46.5559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{271440} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 271440,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.320453006\)
\(L(\frac12)\) \(\approx\) \(2.320453006\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 - T \)
29 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.46248639811768, −12.35330997371298, −11.92532795198686, −11.37969291255362, −10.80173089046091, −10.73393071412101, −10.07154980767818, −9.558469154236148, −9.089107146460321, −8.277530853704496, −8.027732060767326, −7.862997281261081, −7.257910593272031, −6.887588094543397, −5.780522385537400, −5.530330502512339, −5.363396353820962, −4.539699801687741, −4.173368593545174, −3.476813667683055, −3.089986016291792, −2.231057393933717, −1.798421339952703, −1.176602212508807, −0.4272591312140040, 0.4272591312140040, 1.176602212508807, 1.798421339952703, 2.231057393933717, 3.089986016291792, 3.476813667683055, 4.173368593545174, 4.539699801687741, 5.363396353820962, 5.530330502512339, 5.780522385537400, 6.887588094543397, 7.257910593272031, 7.862997281261081, 8.027732060767326, 8.277530853704496, 9.089107146460321, 9.558469154236148, 10.07154980767818, 10.73393071412101, 10.80173089046091, 11.37969291255362, 11.92532795198686, 12.35330997371298, 12.46248639811768

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