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 + 9-s + 5.65·11-s − 5.65·13-s + 5.65·17-s + 4·19-s − 5.65·23-s − 5·25-s − 27-s + 6·29-s − 8·31-s − 5.65·33-s − 2·37-s + 5.65·39-s − 5.65·41-s − 8·47-s − 5.65·51-s + 2·53-s − 4·57-s + 4·59-s − 5.65·61-s − 11.3·67-s + 5.65·69-s + 5.65·71-s − 11.3·73-s + 5·75-s + 11.3·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.333·9-s + 1.70·11-s − 1.56·13-s + 1.37·17-s + 0.917·19-s − 1.17·23-s − 25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.984·33-s − 0.328·37-s + 0.905·39-s − 0.883·41-s − 1.16·47-s − 0.792·51-s + 0.274·53-s − 0.529·57-s + 0.520·59-s − 0.724·61-s − 1.38·67-s + 0.681·69-s + 0.671·71-s − 1.32·73-s + 0.577·75-s + 1.27·79-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 \( 1 + 5T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 5.65T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 5.65T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−7.38763552863504066914788983011, −6.66180757883093863062879148413, −5.97861324327630514907772698902, −5.34404050852531286507562869394, −4.64992658910754387744825163086, −3.83287944982740122886710933174, −3.20394493316144183179453579989, −1.97285604311821539407399239925, −1.23677812021589812783982408537, 0, 1.23677812021589812783982408537, 1.97285604311821539407399239925, 3.20394493316144183179453579989, 3.83287944982740122886710933174, 4.64992658910754387744825163086, 5.34404050852531286507562869394, 5.97861324327630514907772698902, 6.66180757883093863062879148413, 7.38763552863504066914788983011

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