Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7 $
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 + 7-s + 9-s + 6·11-s − 2·13-s + 4·19-s − 21-s − 6·23-s − 5·25-s − 27-s − 6·29-s + 8·31-s − 6·33-s − 2·37-s + 2·39-s + 12·41-s + 4·43-s + 12·47-s + 49-s + 6·53-s − 4·57-s + 10·61-s + 63-s − 8·67-s + 6·69-s + 6·71-s − 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.917·19-s − 0.218·21-s − 1.25·23-s − 25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 1.04·33-s − 0.328·37-s + 0.320·39-s + 1.87·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 0.529·57-s + 1.28·61-s + 0.125·63-s − 0.977·67-s + 0.722·69-s + 0.712·71-s − 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1344} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1344,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.548788080\)
\(L(\frac12)\)  \(\approx\)  \(1.548788080\)
\(L(\frac{3}{2})\)   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 \( 1 - T \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−9.583604842254255789237140171162, −9.000844289389290993593497586164, −7.82437442960794093330464733075, −7.19234324757373870654113009154, −6.18151745230772575654613193139, −5.60383328043951209454966346428, −4.37831328158321326146946665271, −3.81005665439346412696166766304, −2.20954050006078144234673698125, −0.985503276739119959400631494839, 0.985503276739119959400631494839, 2.20954050006078144234673698125, 3.81005665439346412696166766304, 4.37831328158321326146946665271, 5.60383328043951209454966346428, 6.18151745230772575654613193139, 7.19234324757373870654113009154, 7.82437442960794093330464733075, 9.000844289389290993593497586164, 9.583604842254255789237140171162

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