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Properties

Label 2-221760-1.1-c1-0-6
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 − 6·23-s + 25-s − 2·29-s + 6·31-s + 35-s − 2·37-s − 8·43-s − 4·47-s + 49-s + 6·53-s − 55-s − 6·59-s − 8·61-s + 2·65-s + 10·67-s + 6·73-s − 77-s − 16·79-s + 12·83-s + 2·85-s + 10·89-s + 2·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 0.301·11-s − 0.554·13-s − 0.485·17-s − 1.25·23-s + 1/5·25-s − 0.371·29-s + 1.07·31-s + 0.169·35-s − 0.328·37-s − 1.21·43-s − 0.583·47-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 + 1.22·67-s + 0.702·73-s − 0.113·77-s − 1.80·79-s + 1.31·83-s + 0.216·85-s + 1.05·89-s + 0.209·91-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

\(L(1)\) \(\approx\) \(0.5759655629\)
\(L(\frac12)\) \(\approx\) \(0.5759655629\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−12.95637124696706, −12.35618355818682, −12.09331436677939, −11.61641713619162, −11.21839658587000, −10.54380896241707, −10.20395954446039, −9.618695841024760, −9.369860846581882, −8.585776183981127, −8.308176568222228, −7.770441861383089, −7.253363951130050, −6.707374980985858, −6.354401114262420, −5.794221148306144, −5.137899619818664, −4.652885297509961, −4.121730599253200, −3.625801381644422, −3.067546033191570, −2.424110306667904, −1.868136756686504, −1.127372977139241, −0.2202784956775327, 0.2202784956775327, 1.127372977139241, 1.868136756686504, 2.424110306667904, 3.067546033191570, 3.625801381644422, 4.121730599253200, 4.652885297509961, 5.137899619818664, 5.794221148306144, 6.354401114262420, 6.707374980985858, 7.253363951130050, 7.770441861383089, 8.308176568222228, 8.585776183981127, 9.369860846581882, 9.618695841024760, 10.20395954446039, 10.54380896241707, 11.21839658587000, 11.61641713619162, 12.09331436677939, 12.35618355818682, 12.95637124696706

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