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Properties

Label 2-221760-1.1-c1-0-336
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s + 11-s + 6·13-s + 7·17-s + 5·19-s + 23-s + 25-s − 5·29-s − 8·31-s + 35-s + 2·37-s − 12·41-s + 11·43-s − 8·47-s + 49-s − 11·53-s + 55-s − 5·59-s − 7·61-s + 6·65-s + 2·67-s − 12·71-s + 4·73-s + 77-s − 10·79-s − 83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.301·11-s + 1.66·13-s + 1.69·17-s + 1.14·19-s + 0.208·23-s + 1/5·25-s − 0.928·29-s − 1.43·31-s + 0.169·35-s + 0.328·37-s − 1.87·41-s + 1.67·43-s − 1.16·47-s + 1/7·49-s − 1.51·53-s + 0.134·55-s − 0.650·59-s − 0.896·61-s + 0.744·65-s + 0.244·67-s − 1.42·71-s + 0.468·73-s + 0.113·77-s − 1.12·79-s − 0.109·83-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: \(1\)
Selberg data: \((2,\ 221760,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−13.25926915323878, −12.75660056594423, −12.28297956641682, −11.74070371286631, −11.32499972573259, −10.86799701016679, −10.53296805357262, −9.742002669299609, −9.552368504393626, −9.014519154706089, −8.502282415241217, −7.970132408733242, −7.543174119091962, −7.087665544789370, −6.397218674276336, −5.877450613429537, −5.544104261228685, −5.139220420279545, −4.371384789234417, −3.765192235756786, −3.249109871019761, −2.993157526934495, −1.847855787781212, −1.416191013854430, −1.117211155195408, 0, 1.117211155195408, 1.416191013854430, 1.847855787781212, 2.993157526934495, 3.249109871019761, 3.765192235756786, 4.371384789234417, 5.139220420279545, 5.544104261228685, 5.877450613429537, 6.397218674276336, 7.087665544789370, 7.543174119091962, 7.970132408733242, 8.502282415241217, 9.014519154706089, 9.552368504393626, 9.742002669299609, 10.53296805357262, 10.86799701016679, 11.32499972573259, 11.74070371286631, 12.28297956641682, 12.75660056594423, 13.25926915323878

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