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Properties

Label 2-221760-1.1-c1-0-190
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 + 2·13-s + 2·17-s + 25-s − 2·29-s − 4·31-s + 35-s − 6·37-s − 6·41-s − 12·43-s − 4·47-s + 49-s + 6·53-s − 55-s − 4·59-s + 10·61-s − 2·65-s + 4·67-s − 8·71-s − 2·73-s − 77-s − 16·83-s − 2·85-s + 6·89-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 + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.169·35-s − 0.986·37-s − 0.937·41-s − 1.82·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s − 0.134·55-s − 0.520·59-s + 1.28·61-s − 0.248·65-s + 0.488·67-s − 0.949·71-s − 0.234·73-s − 0.113·77-s − 1.75·83-s − 0.216·85-s + 0.635·89-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.11094435191016, −12.83555678522825, −12.19938419734648, −11.73694521044720, −11.47288733640501, −10.92511566459843, −10.23812461045115, −10.09710244403215, −9.450052970728275, −8.887193847979959, −8.471732267424206, −8.146231922117749, −7.348373939037698, −7.088895358651618, −6.533664056286403, −6.054824902631758, −5.359313045260404, −5.104519119452039, −4.278878723009969, −3.828686323110508, −3.293766102038311, −2.983464470460985, −1.931518047702432, −1.594382706023432, −0.7147860619458413, 0, 0.7147860619458413, 1.594382706023432, 1.931518047702432, 2.983464470460985, 3.293766102038311, 3.828686323110508, 4.278878723009969, 5.104519119452039, 5.359313045260404, 6.054824902631758, 6.533664056286403, 7.088895358651618, 7.348373939037698, 8.146231922117749, 8.471732267424206, 8.887193847979959, 9.450052970728275, 10.09710244403215, 10.23812461045115, 10.92511566459843, 11.47288733640501, 11.73694521044720, 12.19938419734648, 12.83555678522825, 13.11094435191016

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