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 − 1.41·13-s − 1.41·15-s − 1.41·17-s + 2.82·23-s − 2.99·25-s + 27-s − 4·31-s + 2.82·33-s − 4·37-s − 1.41·39-s + 1.41·41-s + 5.65·43-s − 1.41·45-s − 12·47-s − 1.41·51-s − 10·53-s − 4.00·55-s − 1.41·61-s + 2.00·65-s + 11.3·67-s + 2.82·69-s − 2.82·71-s − 12.7·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.632·5-s + 0.333·9-s + 0.852·11-s − 0.392·13-s − 0.365·15-s − 0.342·17-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 − 0.226·39-s + 0.220·41-s + 0.862·43-s − 0.210·45-s − 1.75·47-s − 0.198·51-s − 1.37·53-s − 0.539·55-s − 0.181·61-s + 0.248·65-s + 1.38·67-s + 0.340·69-s − 0.335·71-s − 1.48·73-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 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 1.41T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 - 9.89T + 97T^{2} \)
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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

−7.39961299145506272372634150237, −6.82273205312770361074092063355, −6.13254126799236570809664307564, −5.15722395037922820715191160760, −4.45486381885996857575382657400, −3.76137653143114479779026594425, −3.17483802462248879729345800131, −2.19219440194281609499012603618, −1.32335561967512417535557346485, 0, 1.32335561967512417535557346485, 2.19219440194281609499012603618, 3.17483802462248879729345800131, 3.76137653143114479779026594425, 4.45486381885996857575382657400, 5.15722395037922820715191160760, 6.13254126799236570809664307564, 6.82273205312770361074092063355, 7.39961299145506272372634150237

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