Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 2·11-s + 5·13-s + 2·17-s + 3·19-s + 2·23-s − 5·25-s + 27-s − 8·29-s + 31-s − 2·33-s + 5·37-s + 5·39-s − 2·41-s − 7·43-s + 8·47-s + 2·51-s + 2·53-s + 3·57-s + 10·59-s − 2·61-s + 11·67-s + 2·69-s + 12·71-s + 3·73-s − 5·75-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.603·11-s + 1.38·13-s + 0.485·17-s + 0.688·19-s + 0.417·23-s − 25-s + 0.192·27-s − 1.48·29-s + 0.179·31-s − 0.348·33-s + 0.821·37-s + 0.800·39-s − 0.312·41-s − 1.06·43-s + 1.16·47-s + 0.280·51-s + 0.274·53-s + 0.397·57-s + 1.30·59-s − 0.256·61-s + 1.34·67-s + 0.240·69-s + 1.42·71-s + 0.351·73-s − 0.577·75-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  =  \(0\)
Selberg data  =  \((2,\ 9408,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.810822638\)
\(L(\frac12)\)  \(\approx\)  \(2.810822638\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.77856230829999934700256918308, −7.17537709405014070126440776220, −6.32521188915037649724471218916, −5.61449507518201803695944200286, −5.05107329138906055790883903559, −3.85935274105636199851587700207, −3.60921480181870096174475569241, −2.63275157407730932757053124506, −1.78207962432952837703916991087, −0.800310661242121224970869786358, 0.800310661242121224970869786358, 1.78207962432952837703916991087, 2.63275157407730932757053124506, 3.60921480181870096174475569241, 3.85935274105636199851587700207, 5.05107329138906055790883903559, 5.61449507518201803695944200286, 6.32521188915037649724471218916, 7.17537709405014070126440776220, 7.77856230829999934700256918308

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