Properties

Label 2-20160-1.1-c1-0-64
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 5·11-s + 5·13-s + 7·17-s + 2·19-s + 2·23-s + 25-s + 7·29-s + 4·31-s + 35-s + 6·37-s + 12·41-s + 2·43-s − 47-s + 49-s − 5·55-s − 4·59-s − 4·61-s + 5·65-s − 8·67-s + 6·73-s − 5·77-s − 3·79-s − 4·83-s + 7·85-s + 5·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.50·11-s + 1.38·13-s + 1.69·17-s + 0.458·19-s + 0.417·23-s + 1/5·25-s + 1.29·29-s + 0.718·31-s + 0.169·35-s + 0.986·37-s + 1.87·41-s + 0.304·43-s − 0.145·47-s + 1/7·49-s − 0.674·55-s − 0.520·59-s − 0.512·61-s + 0.620·65-s − 0.977·67-s + 0.702·73-s − 0.569·77-s − 0.337·79-s − 0.439·83-s + 0.759·85-s + 0.524·91-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: \(0\)
Selberg data: \((2,\ 20160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.237848191\)
\(L(\frac12)\) \(\approx\) \(3.237848191\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−15.73593808060138, −15.17341741898694, −14.44846201485236, −13.95902906747386, −13.58524261386504, −12.86268932318989, −12.54889780585635, −11.75478260937970, −11.19313560428539, −10.56120496398044, −10.24113554599349, −9.581789404081353, −8.900723508103308, −8.218012091983444, −7.790536168299867, −7.322548635576148, −6.153587864700317, −5.999524131316417, −5.204734862068180, −4.707383041058693, −3.805852106112432, −2.952171059503247, −2.576696915809177, −1.349291651369208, −0.8433102132721604, 0.8433102132721604, 1.349291651369208, 2.576696915809177, 2.952171059503247, 3.805852106112432, 4.707383041058693, 5.204734862068180, 5.999524131316417, 6.153587864700317, 7.322548635576148, 7.790536168299867, 8.218012091983444, 8.900723508103308, 9.581789404081353, 10.24113554599349, 10.56120496398044, 11.19313560428539, 11.75478260937970, 12.54889780585635, 12.86268932318989, 13.58524261386504, 13.95902906747386, 14.44846201485236, 15.17341741898694, 15.73593808060138

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