Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5 \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 7-s + 9-s − 2·13-s + 15-s − 6·17-s − 8·19-s − 21-s + 25-s − 27-s − 6·29-s − 4·31-s − 35-s + 10·37-s + 2·39-s − 6·41-s + 4·43-s − 45-s + 49-s + 6·51-s + 6·53-s + 8·57-s + 12·59-s + 10·61-s + 63-s + 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 1.45·17-s − 1.83·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.169·35-s + 1.64·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.840·51-s + 0.824·53-s + 1.05·57-s + 1.56·59-s + 1.28·61-s + 0.125·63-s + 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6720\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6720} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6720,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.8320264392\)
\(L(\frac12)\)  \(\approx\)  \(0.8320264392\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
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 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.23547869485714, −16.74850814130897, −16.14882483050813, −15.43518539010783, −14.83306915406772, −14.63865696449935, −13.47830452657676, −12.99539873472816, −12.54009926296272, −11.72352841933719, −11.11723081935150, −10.89048552634491, −10.02339863712133, −9.295919233545851, −8.553903914694481, −8.045677196014288, −7.066103340103452, −6.748386455014798, −5.834366028640385, −5.139689651140043, −4.241353840276873, −3.995432845950065, −2.533205198979792, −1.901257096611856, −0.4630202100369112, 0.4630202100369112, 1.901257096611856, 2.533205198979792, 3.995432845950065, 4.241353840276873, 5.139689651140043, 5.834366028640385, 6.748386455014798, 7.066103340103452, 8.045677196014288, 8.553903914694481, 9.295919233545851, 10.02339863712133, 10.89048552634491, 11.11723081935150, 11.72352841933719, 12.54009926296272, 12.99539873472816, 13.47830452657676, 14.63865696449935, 14.83306915406772, 15.43518539010783, 16.14882483050813, 16.74850814130897, 17.23547869485714

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