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 − 3·5-s + 9-s − 3·11-s + 4·13-s + 3·15-s + 4·19-s + 4·25-s − 27-s − 9·29-s − 31-s + 3·33-s − 8·37-s − 4·39-s + 10·43-s − 3·45-s − 6·47-s + 3·53-s + 9·55-s − 4·57-s − 3·59-s + 10·61-s − 12·65-s + 10·67-s − 6·71-s + 2·73-s − 4·75-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.904·11-s + 1.10·13-s + 0.774·15-s + 0.917·19-s + 4/5·25-s − 0.192·27-s − 1.67·29-s − 0.179·31-s + 0.522·33-s − 1.31·37-s − 0.640·39-s + 1.52·43-s − 0.447·45-s − 0.875·47-s + 0.412·53-s + 1.21·55-s − 0.529·57-s − 0.390·59-s + 1.28·61-s − 1.48·65-s + 1.22·67-s − 0.712·71-s + 0.234·73-s − 0.461·75-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 \( 1 + 3 T + p T^{2} \)
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 \( 1 + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−7.44917126886878782778628212173, −6.81705405211828332190741708071, −5.85646054201011325385165321285, −5.35433498635568884591889300064, −4.58630872717760998840543324315, −3.68808112594794107826379113766, −3.40590973984036055091876086457, −2.13017991921161551159104262118, −0.959115488418008646173488227950, 0, 0.959115488418008646173488227950, 2.13017991921161551159104262118, 3.40590973984036055091876086457, 3.68808112594794107826379113766, 4.58630872717760998840543324315, 5.35433498635568884591889300064, 5.85646054201011325385165321285, 6.81705405211828332190741708071, 7.44917126886878782778628212173

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