Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s + 9-s − 2·11-s − 4·13-s + 15-s + 2·17-s + 2·19-s + 21-s + 4·23-s + 25-s − 27-s + 2·29-s − 6·31-s + 2·33-s + 35-s + 6·37-s + 4·39-s + 6·41-s + 4·43-s − 45-s + 49-s − 2·51-s − 8·53-s + 2·55-s − 2·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.258·15-s + 0.485·17-s + 0.458·19-s + 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.07·31-s + 0.348·33-s + 0.169·35-s + 0.986·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.280·51-s − 1.09·53-s + 0.269·55-s − 0.264·57-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  =  1
Selberg data  =  $(2,\ 6720,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−17.34804958417083, −17.06878441603382, −16.23320671287031, −15.97461650236211, −15.26878946263251, −14.58039622153206, −14.15934097892130, −13.09116126327156, −12.73424133625783, −12.27384089707447, −11.39011315608373, −11.11163991480938, −10.20143695487065, −9.760586056660992, −9.083144771649225, −8.188838164880507, −7.393207762341332, −7.170271217218766, −6.159824013494125, −5.456729495825204, −4.864720473203335, −4.087969475903419, −3.129794904872458, −2.415116248499065, −1.059847187479721, 0, 1.059847187479721, 2.415116248499065, 3.129794904872458, 4.087969475903419, 4.864720473203335, 5.456729495825204, 6.159824013494125, 7.170271217218766, 7.393207762341332, 8.188838164880507, 9.083144771649225, 9.760586056660992, 10.20143695487065, 11.11163991480938, 11.39011315608373, 12.27384089707447, 12.73424133625783, 13.09116126327156, 14.15934097892130, 14.58039622153206, 15.26878946263251, 15.97461650236211, 16.23320671287031, 17.06878441603382, 17.34804958417083

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