Properties

Label 2-52800-1.1-c1-0-151
Degree $2$
Conductor $52800$
Sign $-1$
Analytic cond. $421.610$
Root an. cond. $20.5331$
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 + 2·7-s + 9-s + 11-s + 2·13-s + 2·19-s − 2·21-s − 27-s − 8·31-s − 33-s + 2·37-s − 2·39-s − 2·43-s − 3·49-s + 6·53-s − 2·57-s − 12·59-s − 2·61-s + 2·63-s + 4·67-s − 2·73-s + 2·77-s + 10·79-s + 81-s + 12·83-s − 6·89-s + 4·91-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.458·19-s − 0.436·21-s − 0.192·27-s − 1.43·31-s − 0.174·33-s + 0.328·37-s − 0.320·39-s − 0.304·43-s − 3/7·49-s + 0.824·53-s − 0.264·57-s − 1.56·59-s − 0.256·61-s + 0.251·63-s + 0.488·67-s − 0.234·73-s + 0.227·77-s + 1.12·79-s + 1/9·81-s + 1.31·83-s − 0.635·89-s + 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52800\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(421.610\)
Root analytic conductor: \(20.5331\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 52800,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
11 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.81145488418416, −14.13503304934052, −13.75341110908278, −13.17945817761252, −12.57042704352489, −12.16422709794272, −11.51387566909507, −11.18234914139282, −10.75735862049049, −10.17077633527507, −9.499448168559340, −9.048299867950435, −8.458823500866258, −7.788280923500972, −7.432142476673968, −6.694513255927098, −6.206175356123614, −5.559569811860627, −5.096521408307136, −4.505264304888195, −3.827798936620099, −3.295435268389742, −2.326950170444910, −1.600164487303859, −1.041587016101660, 0, 1.041587016101660, 1.600164487303859, 2.326950170444910, 3.295435268389742, 3.827798936620099, 4.505264304888195, 5.096521408307136, 5.559569811860627, 6.206175356123614, 6.694513255927098, 7.432142476673968, 7.788280923500972, 8.458823500866258, 9.048299867950435, 9.499448168559340, 10.17077633527507, 10.75735862049049, 11.18234914139282, 11.51387566909507, 12.16422709794272, 12.57042704352489, 13.17945817761252, 13.75341110908278, 14.13503304934052, 14.81145488418416

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