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 + 1.04·5-s + 9-s − 5.93·11-s + 0.888·13-s − 1.04·15-s + 4.51·17-s − 3.47·19-s − 1.93·23-s − 3.91·25-s − 27-s + 8.38·29-s + 5.13·31-s + 5.93·33-s − 5.65·37-s − 0.888·39-s + 8.39·41-s + 0.743·43-s + 1.04·45-s − 4.21·47-s − 4.51·51-s + 6.91·53-s − 6.18·55-s + 3.47·57-s − 5.43·59-s − 10.9·61-s + 0.926·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.466·5-s + 0.333·9-s − 1.78·11-s + 0.246·13-s − 0.269·15-s + 1.09·17-s − 0.797·19-s − 0.402·23-s − 0.782·25-s − 0.192·27-s + 1.55·29-s + 0.921·31-s + 1.03·33-s − 0.929·37-s − 0.142·39-s + 1.31·41-s + 0.113·43-s + 0.155·45-s − 0.615·47-s − 0.632·51-s + 0.949·53-s − 0.833·55-s + 0.460·57-s − 0.708·59-s − 1.40·61-s + 0.114·65-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 \( 1 - 1.04T + 5T^{2} \)
11 \( 1 + 5.93T + 11T^{2} \)
13 \( 1 - 0.888T + 13T^{2} \)
17 \( 1 - 4.51T + 17T^{2} \)
19 \( 1 + 3.47T + 19T^{2} \)
23 \( 1 + 1.93T + 23T^{2} \)
29 \( 1 - 8.38T + 29T^{2} \)
31 \( 1 - 5.13T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 - 8.39T + 41T^{2} \)
43 \( 1 - 0.743T + 43T^{2} \)
47 \( 1 + 4.21T + 47T^{2} \)
53 \( 1 - 6.91T + 53T^{2} \)
59 \( 1 + 5.43T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 - 6.20T + 67T^{2} \)
71 \( 1 - 3.72T + 71T^{2} \)
73 \( 1 + 3.49T + 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 - 2.94T + 83T^{2} \)
89 \( 1 + 5.00T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−7.46462825475087322114646619717, −6.45954050599013324834277821361, −6.01271147438861012655639884049, −5.30270751690245038332371245862, −4.80334406988242374780926916274, −3.90162253843110061224610998685, −2.88540342648179512321213156606, −2.24761283559428606605929140686, −1.13836171604517499477148773315, 0, 1.13836171604517499477148773315, 2.24761283559428606605929140686, 2.88540342648179512321213156606, 3.90162253843110061224610998685, 4.80334406988242374780926916274, 5.30270751690245038332371245862, 6.01271147438861012655639884049, 6.45954050599013324834277821361, 7.46462825475087322114646619717

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