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Properties

Label 2-20160-1.1-c1-0-85
Degree $2$
Conductor $20160$
Sign $-1$
Analytic cond. $160.978$
Root an. cond. $12.6877$
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 − 2·13-s + 6·17-s − 4·19-s + 25-s − 6·29-s + 4·31-s + 35-s − 2·37-s − 6·41-s + 8·43-s − 12·47-s + 49-s + 6·53-s + 12·59-s − 2·61-s + 2·65-s + 8·67-s + 14·73-s + 16·79-s − 12·83-s − 6·85-s − 6·89-s + 2·91-s + 4·95-s + 14·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.554·13-s + 1.45·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.169·35-s − 0.328·37-s − 0.937·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s + 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.977·67-s + 1.63·73-s + 1.80·79-s − 1.31·83-s − 0.650·85-s − 0.635·89-s + 0.209·91-s + 0.410·95-s + 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20160\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(160.978\)
Root analytic conductor: \(12.6877\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 20160,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.87287776429853, −15.41299095883298, −14.70731861107425, −14.51097953930977, −13.77545276707056, −12.99884994086245, −12.75713266914218, −11.93919280523442, −11.77571198358923, −10.86681270465987, −10.43300995510210, −9.711007201979599, −9.418154341405471, −8.456429654295259, −8.093246397827276, −7.441478880680266, −6.819512721055411, −6.264890125232296, −5.401306389690488, −5.019126364888597, −4.006931016319367, −3.619449268033432, −2.791842450581719, −2.031679784750651, −0.9886551163763003, 0, 0.9886551163763003, 2.031679784750651, 2.791842450581719, 3.619449268033432, 4.006931016319367, 5.019126364888597, 5.401306389690488, 6.264890125232296, 6.819512721055411, 7.441478880680266, 8.093246397827276, 8.456429654295259, 9.418154341405471, 9.711007201979599, 10.43300995510210, 10.86681270465987, 11.77571198358923, 11.93919280523442, 12.75713266914218, 12.99884994086245, 13.77545276707056, 14.51097953930977, 14.70731861107425, 15.41299095883298, 15.87287776429853

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