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 − 3.20·5-s + 9-s − 4.24·11-s − 3.15·13-s − 3.20·15-s + 4.40·17-s + 3.15·19-s + 4.40·23-s + 5.24·25-s + 27-s − 7.20·29-s − 2.04·31-s − 4.24·33-s + 9.65·37-s − 3.15·39-s + 10.4·41-s − 0.750·43-s − 3.20·45-s + 2.40·47-s + 4.40·51-s + 3.29·53-s + 13.6·55-s + 3.15·57-s + 8.24·59-s − 8.09·61-s + 10.0·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.43·5-s + 0.333·9-s − 1.28·11-s − 0.874·13-s − 0.826·15-s + 1.06·17-s + 0.723·19-s + 0.918·23-s + 1.04·25-s + 0.192·27-s − 1.33·29-s − 0.367·31-s − 0.739·33-s + 1.58·37-s − 0.504·39-s + 1.63·41-s − 0.114·43-s − 0.477·45-s + 0.350·47-s + 0.616·51-s + 0.452·53-s + 1.83·55-s + 0.417·57-s + 1.07·59-s − 1.03·61-s + 1.25·65-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 \( 1 + 3.20T + 5T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 + 3.15T + 13T^{2} \)
17 \( 1 - 4.40T + 17T^{2} \)
19 \( 1 - 3.15T + 19T^{2} \)
23 \( 1 - 4.40T + 23T^{2} \)
29 \( 1 + 7.20T + 29T^{2} \)
31 \( 1 + 2.04T + 31T^{2} \)
37 \( 1 - 9.65T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 0.750T + 43T^{2} \)
47 \( 1 - 2.40T + 47T^{2} \)
53 \( 1 - 3.29T + 53T^{2} \)
59 \( 1 - 8.24T + 59T^{2} \)
61 \( 1 + 8.09T + 61T^{2} \)
67 \( 1 + 5.15T + 67T^{2} \)
71 \( 1 - 6.40T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + 16.4T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + 2.40T + 89T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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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

−7.48073371779343684785431177529, −7.20405945165639230519978538402, −5.84180859648746657195865322276, −5.23054137436340224665059310295, −4.46217330808684756169947671372, −3.78719663386081581360766789617, −2.99588917665941883418342124759, −2.50756424549314561971395587214, −1.09315840968816290695441560999, 0, 1.09315840968816290695441560999, 2.50756424549314561971395587214, 2.99588917665941883418342124759, 3.78719663386081581360766789617, 4.46217330808684756169947671372, 5.23054137436340224665059310295, 5.84180859648746657195865322276, 7.20405945165639230519978538402, 7.48073371779343684785431177529

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