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Properties

Label 2-221760-1.1-c1-0-318
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 + 6·17-s − 4·19-s + 8·23-s + 25-s + 6·29-s − 8·31-s + 35-s − 6·37-s + 6·41-s − 4·43-s − 8·47-s + 49-s − 10·53-s − 55-s + 12·59-s + 10·61-s + 2·65-s − 12·67-s − 8·71-s − 6·73-s − 77-s − 8·79-s + 4·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.169·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 1.37·53-s − 0.134·55-s + 1.56·59-s + 1.28·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s − 0.702·73-s − 0.113·77-s − 0.900·79-s + 0.439·83-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−13.17581955886562, −12.70428965547193, −12.44535357006516, −11.75824193912638, −11.23941407591036, −10.89725847423050, −10.42149399163681, −9.940132327453319, −9.549137742681822, −8.795117071329530, −8.575717760825920, −8.121999580413890, −7.392383450907388, −7.074412659630046, −6.521452979500095, −5.870866402726736, −5.519990189772240, −4.984611221296656, −4.534484294679515, −3.818841192163398, −3.184059245964033, −2.873994989312525, −2.018526661426993, −1.455991013922759, −0.9663732376931495, 0, 0.9663732376931495, 1.455991013922759, 2.018526661426993, 2.873994989312525, 3.184059245964033, 3.818841192163398, 4.534484294679515, 4.984611221296656, 5.519990189772240, 5.870866402726736, 6.521452979500095, 7.074412659630046, 7.392383450907388, 8.121999580413890, 8.575717760825920, 8.795117071329530, 9.549137742681822, 9.940132327453319, 10.42149399163681, 10.89725847423050, 11.23941407591036, 11.75824193912638, 12.44535357006516, 12.70428965547193, 13.17581955886562

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