Properties

Degree $2$
Conductor $20160$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 2·13-s + 2·19-s + 25-s + 6·29-s − 8·31-s + 35-s + 4·37-s − 6·41-s + 2·43-s − 6·47-s + 49-s + 6·53-s − 12·59-s − 8·61-s + 2·65-s + 2·67-s + 6·71-s + 2·73-s + 16·79-s − 6·89-s + 2·91-s − 2·95-s − 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.554·13-s + 0.458·19-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.169·35-s + 0.657·37-s − 0.937·41-s + 0.304·43-s − 0.875·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s − 1.02·61-s + 0.248·65-s + 0.244·67-s + 0.712·71-s + 0.234·73-s + 1.80·79-s − 0.635·89-s + 0.209·91-s − 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-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: $1$
Motivic weight: \(1\)
Character: $\chi_{20160} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 20160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.302950348\)
\(L(\frac12)\) \(\approx\) \(1.302950348\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.72944154389818, −14.98218471283237, −14.78011284311924, −13.89051358223114, −13.60917684759037, −12.80342258765582, −12.31448886759987, −11.96524927584644, −11.16792496072321, −10.77769822216359, −10.03052025221888, −9.536778878279940, −8.987914699678905, −8.269274799877019, −7.720075036284779, −7.137356726879416, −6.563575196796162, −5.884860654758734, −5.114034251577523, −4.608060960531428, −3.767894899086732, −3.190122837814920, −2.461257913590594, −1.519671559269586, −0.4740798824174354, 0.4740798824174354, 1.519671559269586, 2.461257913590594, 3.190122837814920, 3.767894899086732, 4.608060960531428, 5.114034251577523, 5.884860654758734, 6.563575196796162, 7.137356726879416, 7.720075036284779, 8.269274799877019, 8.987914699678905, 9.536778878279940, 10.03052025221888, 10.77769822216359, 11.16792496072321, 11.96524927584644, 12.31448886759987, 12.80342258765582, 13.60917684759037, 13.89051358223114, 14.78011284311924, 14.98218471283237, 15.72944154389818

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