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Properties

Label 2-221760-1.1-c1-0-286
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 − 4·13-s + 4·17-s + 8·19-s + 2·23-s + 25-s − 6·29-s + 4·31-s − 35-s − 4·37-s − 2·41-s + 4·43-s − 4·47-s + 49-s + 2·53-s + 55-s − 10·61-s − 4·65-s + 14·67-s + 8·71-s − 10·73-s − 77-s − 4·79-s + 4·83-s + 4·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.301·11-s − 1.10·13-s + 0.970·17-s + 1.83·19-s + 0.417·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.169·35-s − 0.657·37-s − 0.312·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.274·53-s + 0.134·55-s − 1.28·61-s − 0.496·65-s + 1.71·67-s + 0.949·71-s − 1.17·73-s − 0.113·77-s − 0.450·79-s + 0.439·83-s + 0.433·85-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 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−13.25852093498316, −12.62863316871996, −12.23104341910571, −11.97415829805909, −11.29284286483109, −10.98804350424583, −10.10644825229510, −9.946823911060292, −9.536107073899149, −9.149310291548892, −8.539767268327385, −7.826371843363958, −7.542865139681782, −6.977832975173504, −6.640025558086328, −5.831567941761840, −5.453592316307568, −5.129824848606117, −4.466539357086276, −3.784281847606962, −3.171819760387344, −2.876362876614080, −2.128097483334833, −1.423469264948573, −0.8898201518113129, 0, 0.8898201518113129, 1.423469264948573, 2.128097483334833, 2.876362876614080, 3.171819760387344, 3.784281847606962, 4.466539357086276, 5.129824848606117, 5.453592316307568, 5.831567941761840, 6.640025558086328, 6.977832975173504, 7.542865139681782, 7.826371843363958, 8.539767268327385, 9.149310291548892, 9.536107073899149, 9.946823911060292, 10.10644825229510, 10.98804350424583, 11.29284286483109, 11.97415829805909, 12.23104341910571, 12.62863316871996, 13.25852093498316

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