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Properties

Label 2-221760-1.1-c1-0-44
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 + 6·17-s + 8·19-s − 6·23-s + 25-s + 6·29-s − 2·31-s + 35-s − 2·37-s + 8·43-s − 12·47-s + 49-s + 6·53-s − 55-s − 6·59-s − 8·61-s + 2·65-s + 2·67-s − 10·73-s − 77-s − 8·79-s − 12·83-s − 6·85-s − 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 0.301·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.359·31-s + 0.169·35-s − 0.328·37-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s − 0.134·55-s − 0.781·59-s − 1.02·61-s + 0.248·65-s + 0.244·67-s − 1.17·73-s − 0.113·77-s − 0.900·79-s − 1.31·83-s − 0.650·85-s − 0.635·89-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

\(L(1)\) \(\approx\) \(1.894725472\)
\(L(\frac12)\) \(\approx\) \(1.894725472\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−12.77096982485972, −12.42206680214487, −12.08218010854778, −11.52296530415293, −11.42187208977898, −10.43037490411576, −10.05904930370926, −9.877153262344744, −9.222360014860905, −8.803939788666301, −8.064213830312505, −7.767549304164185, −7.300066521485393, −6.917541626593620, −6.085448224401658, −5.829027315649352, −5.214172547709810, −4.692090404835277, −4.118037299974092, −3.464760881107008, −3.122765618068257, −2.610367880194992, −1.658392229248809, −1.154382759639231, −0.4150334918923534, 0.4150334918923534, 1.154382759639231, 1.658392229248809, 2.610367880194992, 3.122765618068257, 3.464760881107008, 4.118037299974092, 4.692090404835277, 5.214172547709810, 5.829027315649352, 6.085448224401658, 6.917541626593620, 7.300066521485393, 7.767549304164185, 8.064213830312505, 8.803939788666301, 9.222360014860905, 9.877153262344744, 10.05904930370926, 10.43037490411576, 11.42187208977898, 11.52296530415293, 12.08218010854778, 12.42206680214487, 12.77096982485972

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