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Properties

Label 2-221760-1.1-c1-0-59
Degree $2$
Conductor $221760$
Sign $1$
Analytic cond. $1770.76$
Root an. cond. $42.0804$
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 − 11-s + 2·13-s − 2·17-s − 4·19-s + 25-s + 6·29-s + 35-s − 6·37-s + 6·41-s + 4·43-s + 49-s − 2·53-s − 55-s + 4·59-s − 6·61-s + 2·65-s − 12·67-s + 10·73-s − 77-s + 8·79-s − 4·83-s − 2·85-s − 10·89-s + 2·91-s − 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s + 1.11·29-s + 0.169·35-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.274·53-s − 0.134·55-s + 0.520·59-s − 0.768·61-s + 0.248·65-s − 1.46·67-s + 1.17·73-s − 0.113·77-s + 0.900·79-s − 0.439·83-s − 0.216·85-s − 1.05·89-s + 0.209·91-s − 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221760\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(1770.76\)
Root analytic conductor: \(42.0804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 221760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.562429080\)
\(L(\frac12)\) \(\approx\) \(2.562429080\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−12.81946733362846, −12.62735016324511, −12.10942534068964, −11.48561984888875, −11.04259422276533, −10.61409096399927, −10.30987127396553, −9.699832499077821, −9.145453744705232, −8.652138923996678, −8.418595242071990, −7.743735489207237, −7.286041552547778, −6.665600162649122, −6.205415515960141, −5.860833950966904, −5.138799474990904, −4.723741757179371, −4.199431018005588, −3.639732547365561, −2.906182858118475, −2.408697044123216, −1.835437296491186, −1.212469010876424, −0.4538934921150039, 0.4538934921150039, 1.212469010876424, 1.835437296491186, 2.408697044123216, 2.906182858118475, 3.639732547365561, 4.199431018005588, 4.723741757179371, 5.138799474990904, 5.860833950966904, 6.205415515960141, 6.665600162649122, 7.286041552547778, 7.743735489207237, 8.418595242071990, 8.652138923996678, 9.145453744705232, 9.699832499077821, 10.30987127396553, 10.61409096399927, 11.04259422276533, 11.48561984888875, 12.10942534068964, 12.62735016324511, 12.81946733362846

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