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 − 1.41·5-s + 9-s − 2.82·11-s + 4.24·13-s − 1.41·15-s + 4.24·17-s − 8·19-s − 2.82·23-s − 2.99·25-s + 27-s + 4·31-s − 2.82·33-s + 4·37-s + 4.24·39-s + 7.07·41-s − 5.65·43-s − 1.41·45-s − 4·47-s + 4.24·51-s + 6·53-s + 4.00·55-s − 8·57-s − 8·59-s + 4.24·61-s − 6·65-s − 2.82·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.632·5-s + 0.333·9-s − 0.852·11-s + 1.17·13-s − 0.365·15-s + 1.02·17-s − 1.83·19-s − 0.589·23-s − 0.599·25-s + 0.192·27-s + 0.718·31-s − 0.492·33-s + 0.657·37-s + 0.679·39-s + 1.10·41-s − 0.862·43-s − 0.210·45-s − 0.583·47-s + 0.594·51-s + 0.824·53-s + 0.539·55-s − 1.05·57-s − 1.04·59-s + 0.543·61-s − 0.744·65-s − 0.340·69-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 \( 1 + 1.41T + 5T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 4.24T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + 1.41T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 1.41T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−7.60706701343412120187420858530, −6.67707781267495867352390317626, −6.06254060858610539765519137491, −5.32657740901729022209059391577, −4.25099612002853895132000788890, −3.94866147183485483451458537503, −3.04388701822641046890994418298, −2.29248202909747841308163648409, −1.27530079570322944130771380821, 0, 1.27530079570322944130771380821, 2.29248202909747841308163648409, 3.04388701822641046890994418298, 3.94866147183485483451458537503, 4.25099612002853895132000788890, 5.32657740901729022209059391577, 6.06254060858610539765519137491, 6.67707781267495867352390317626, 7.60706701343412120187420858530

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