Properties

Degree 2
Conductor $ 2^{4} \cdot 7 \cdot 19 \cdot 47 $
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 − 5-s + 7-s − 2·9-s − 4·11-s − 4·13-s + 15-s + 4·17-s + 19-s − 21-s − 4·23-s − 4·25-s + 5·27-s − 3·29-s − 2·31-s + 4·33-s − 35-s + 8·37-s + 4·39-s − 2·41-s + 8·43-s + 2·45-s − 47-s + 49-s − 4·51-s + 8·53-s + 4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.20·11-s − 1.10·13-s + 0.258·15-s + 0.970·17-s + 0.229·19-s − 0.218·21-s − 0.834·23-s − 4/5·25-s + 0.962·27-s − 0.557·29-s − 0.359·31-s + 0.696·33-s − 0.169·35-s + 1.31·37-s + 0.640·39-s − 0.312·41-s + 1.21·43-s + 0.298·45-s − 0.145·47-s + 1/7·49-s − 0.560·51-s + 1.09·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100016\)    =    \(2^{4} \cdot 7 \cdot 19 \cdot 47\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100016} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 100016,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;19,\;47\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;19,\;47\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 - T \)
19 \( 1 - T \)
47 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−14.19045401658049, −13.44764790345246, −12.99833534336357, −12.32443616392812, −11.95223524737756, −11.76927430563300, −10.95030535297755, −10.72239759371306, −10.15415893781042, −9.567113349674381, −9.173874205706664, −8.303125477815227, −7.864776027022198, −7.636785957385236, −7.119255776495497, −6.219222075266591, −5.643024465334305, −5.518185181625404, −4.749799634232581, −4.333222779065209, −3.548068254138568, −2.848595700892036, −2.423286223159660, −1.616009115708318, −0.6049027137249306, 0, 0.6049027137249306, 1.616009115708318, 2.423286223159660, 2.848595700892036, 3.548068254138568, 4.333222779065209, 4.749799634232581, 5.518185181625404, 5.643024465334305, 6.219222075266591, 7.119255776495497, 7.636785957385236, 7.864776027022198, 8.303125477815227, 9.173874205706664, 9.567113349674381, 10.15415893781042, 10.72239759371306, 10.95030535297755, 11.76927430563300, 11.95223524737756, 12.32443616392812, 12.99833534336357, 13.44764790345246, 14.19045401658049

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