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 − 2.13·5-s + 9-s − 0.298·11-s + 6.43·13-s + 2.13·15-s − 1.11·17-s + 1.01·19-s − 4.29·23-s − 0.441·25-s − 27-s + 0.422·29-s − 10.6·31-s + 0.298·33-s + 5.65·37-s − 6.43·39-s − 8.32·41-s + 7.09·43-s − 2.13·45-s + 8.11·47-s + 1.11·51-s + 3.44·53-s + 0.637·55-s − 1.01·57-s − 6.46·59-s + 5.83·61-s − 13.7·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.954·5-s + 0.333·9-s − 0.0900·11-s + 1.78·13-s + 0.551·15-s − 0.270·17-s + 0.233·19-s − 0.896·23-s − 0.0883·25-s − 0.192·27-s + 0.0784·29-s − 1.91·31-s + 0.0519·33-s + 0.929·37-s − 1.03·39-s − 1.30·41-s + 1.08·43-s − 0.318·45-s + 1.18·47-s + 0.156·51-s + 0.472·53-s + 0.0859·55-s − 0.135·57-s − 0.841·59-s + 0.747·61-s − 1.70·65-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 \( 1 + 2.13T + 5T^{2} \)
11 \( 1 + 0.298T + 11T^{2} \)
13 \( 1 - 6.43T + 13T^{2} \)
17 \( 1 + 1.11T + 17T^{2} \)
19 \( 1 - 1.01T + 19T^{2} \)
23 \( 1 + 4.29T + 23T^{2} \)
29 \( 1 - 0.422T + 29T^{2} \)
31 \( 1 + 10.6T + 31T^{2} \)
37 \( 1 - 5.65T + 37T^{2} \)
41 \( 1 + 8.32T + 41T^{2} \)
43 \( 1 - 7.09T + 43T^{2} \)
47 \( 1 - 8.11T + 47T^{2} \)
53 \( 1 - 3.44T + 53T^{2} \)
59 \( 1 + 6.46T + 59T^{2} \)
61 \( 1 - 5.83T + 61T^{2} \)
67 \( 1 + 5.05T + 67T^{2} \)
71 \( 1 - 1.35T + 71T^{2} \)
73 \( 1 - 2.85T + 73T^{2} \)
79 \( 1 - 7.69T + 79T^{2} \)
83 \( 1 + 6.03T + 83T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 - 0.512T + 97T^{2} \)
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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

−7.39089128613767640470233922663, −6.68207573946043715016309306417, −5.88785384846518602811612397857, −5.50865443828201507117624402298, −4.36788285319638928933365559501, −3.92411821742212657760680762575, −3.30717210351103327513355307926, −2.05017372324685968058761233870, −1.06640451220701885007517204677, 0, 1.06640451220701885007517204677, 2.05017372324685968058761233870, 3.30717210351103327513355307926, 3.92411821742212657760680762575, 4.36788285319638928933365559501, 5.50865443828201507117624402298, 5.88785384846518602811612397857, 6.68207573946043715016309306417, 7.39089128613767640470233922663

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