Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 7-s − 6·11-s − 6·13-s − 2·17-s − 4·19-s − 2·23-s − 25-s + 8·29-s − 4·31-s − 2·35-s − 6·37-s + 10·41-s + 4·43-s + 4·47-s + 49-s − 4·53-s − 12·55-s + 12·59-s − 2·61-s − 12·65-s − 12·67-s − 6·71-s − 2·73-s + 6·77-s + 8·79-s − 4·85-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 1.80·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s − 0.417·23-s − 1/5·25-s + 1.48·29-s − 0.718·31-s − 0.338·35-s − 0.986·37-s + 1.56·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.549·53-s − 1.61·55-s + 1.56·59-s − 0.256·61-s − 1.48·65-s − 1.46·67-s − 0.712·71-s − 0.234·73-s + 0.683·77-s + 0.900·79-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−9.741330569049991606639655841914, −8.799787487059594666120880479048, −7.81574295644308953149169772105, −7.06349862549190184985368579025, −6.01034584523872431338499373875, −5.26501418343618196652528930650, −4.39604700645425257403950589550, −2.73183663384879672157762604102, −2.21321859651013231545979667053, 0, 2.21321859651013231545979667053, 2.73183663384879672157762604102, 4.39604700645425257403950589550, 5.26501418343618196652528930650, 6.01034584523872431338499373875, 7.06349862549190184985368579025, 7.81574295644308953149169772105, 8.799787487059594666120880479048, 9.741330569049991606639655841914

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