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Properties

Label 2-20160-1.1-c1-0-133
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 − 4·11-s + 6·13-s − 6·17-s + 4·19-s + 4·23-s + 25-s − 2·29-s + 8·31-s + 35-s − 6·37-s − 6·41-s − 8·43-s + 49-s + 6·53-s − 4·55-s − 4·59-s − 10·61-s + 6·65-s − 8·67-s − 12·71-s − 14·73-s − 4·77-s − 16·79-s − 12·83-s − 6·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.20·11-s + 1.66·13-s − 1.45·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.169·35-s − 0.986·37-s − 0.937·41-s − 1.21·43-s + 1/7·49-s + 0.824·53-s − 0.539·55-s − 0.520·59-s − 1.28·61-s + 0.744·65-s − 0.977·67-s − 1.42·71-s − 1.63·73-s − 0.455·77-s − 1.80·79-s − 1.31·83-s − 0.650·85-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−15.82010289505940, −15.45536000914478, −15.01972932022900, −14.11032341295887, −13.57771660901576, −13.30042257704873, −12.96229439349979, −11.89776303949978, −11.54118588135015, −10.89107419102308, −10.42918212990362, −9.978717685310743, −9.020135403933464, −8.630545137579536, −8.244213334400963, −7.294556595694272, −6.903200225498636, −6.003800697001181, −5.671698490154158, −4.787849503539347, −4.422862926236138, −3.269088760072350, −2.896194751607939, −1.849919175504944, −1.245724170298772, 0, 1.245724170298772, 1.849919175504944, 2.896194751607939, 3.269088760072350, 4.422862926236138, 4.787849503539347, 5.671698490154158, 6.003800697001181, 6.903200225498636, 7.294556595694272, 8.244213334400963, 8.630545137579536, 9.020135403933464, 9.978717685310743, 10.42918212990362, 10.89107419102308, 11.54118588135015, 11.89776303949978, 12.96229439349979, 13.30042257704873, 13.57771660901576, 14.11032341295887, 15.01972932022900, 15.45536000914478, 15.82010289505940

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