Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7^{2} $
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 + 9-s + 3·11-s − 4·13-s + 15-s − 4·19-s − 8·23-s − 4·25-s + 27-s + 3·29-s + 5·31-s + 3·33-s − 8·37-s − 4·39-s + 8·41-s + 6·43-s + 45-s − 10·47-s − 9·53-s + 3·55-s − 4·57-s − 5·59-s + 10·61-s − 4·65-s + 6·67-s − 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.904·11-s − 1.10·13-s + 0.258·15-s − 0.917·19-s − 1.66·23-s − 4/5·25-s + 0.192·27-s + 0.557·29-s + 0.898·31-s + 0.522·33-s − 1.31·37-s − 0.640·39-s + 1.24·41-s + 0.914·43-s + 0.149·45-s − 1.45·47-s − 1.23·53-s + 0.404·55-s − 0.529·57-s − 0.650·59-s + 1.28·61-s − 0.496·65-s + 0.733·67-s − 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9408} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 9408,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−7.42726162911028705331278008951, −6.60305999567375257133497036704, −6.16627383952669880098720050859, −5.29756157702985957157895239329, −4.36148731296550932502233153823, −3.97688462267904261970972422654, −2.89872148536241795152805641294, −2.18483816931715435376045494244, −1.48235761878362885435682130382, 0, 1.48235761878362885435682130382, 2.18483816931715435376045494244, 2.89872148536241795152805641294, 3.97688462267904261970972422654, 4.36148731296550932502233153823, 5.29756157702985957157895239329, 6.16627383952669880098720050859, 6.60305999567375257133497036704, 7.42726162911028705331278008951

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