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 − 3.04·5-s + 9-s + 3.93·11-s − 4.88·13-s + 3.04·15-s − 5.34·17-s + 2.30·19-s + 7.93·23-s + 4.25·25-s − 27-s − 5.55·29-s − 0.645·31-s − 3.93·33-s − 5.65·37-s + 4.88·39-s + 10.0·41-s + 8.91·43-s − 3.04·45-s − 6.61·47-s + 5.34·51-s − 1.25·53-s − 11.9·55-s − 2.30·57-s − 3.04·59-s + 2.97·61-s + 14.8·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.36·5-s + 0.333·9-s + 1.18·11-s − 1.35·13-s + 0.785·15-s − 1.29·17-s + 0.528·19-s + 1.65·23-s + 0.851·25-s − 0.192·27-s − 1.03·29-s − 0.115·31-s − 0.684·33-s − 0.929·37-s + 0.782·39-s + 1.57·41-s + 1.35·43-s − 0.453·45-s − 0.964·47-s + 0.748·51-s − 0.172·53-s − 1.61·55-s − 0.304·57-s − 0.396·59-s + 0.380·61-s + 1.84·65-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 \( 1 + 3.04T + 5T^{2} \)
11 \( 1 - 3.93T + 11T^{2} \)
13 \( 1 + 4.88T + 13T^{2} \)
17 \( 1 + 5.34T + 17T^{2} \)
19 \( 1 - 2.30T + 19T^{2} \)
23 \( 1 - 7.93T + 23T^{2} \)
29 \( 1 + 5.55T + 29T^{2} \)
31 \( 1 + 0.645T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 - 8.91T + 43T^{2} \)
47 \( 1 + 6.61T + 47T^{2} \)
53 \( 1 + 1.25T + 53T^{2} \)
59 \( 1 + 3.04T + 59T^{2} \)
61 \( 1 - 2.97T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 4.67T + 73T^{2} \)
79 \( 1 - 1.05T + 79T^{2} \)
83 \( 1 + 8.60T + 83T^{2} \)
89 \( 1 - 4.85T + 89T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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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

−7.31338302660946523141574540612, −6.87182812029735496170313309912, −6.08978493908023287700425180013, −5.09112492114489264301380767606, −4.58265447813848175119506277788, −3.95490298626924780592256400373, −3.20429494256637114063498794092, −2.16274980201029654054923278701, −0.955707403670205989187915268043, 0, 0.955707403670205989187915268043, 2.16274980201029654054923278701, 3.20429494256637114063498794092, 3.95490298626924780592256400373, 4.58265447813848175119506277788, 5.09112492114489264301380767606, 6.08978493908023287700425180013, 6.87182812029735496170313309912, 7.31338302660946523141574540612

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