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  − 2·3-s − 2·5-s − 7-s + 9-s + 2·13-s + 4·15-s + 6·17-s − 19-s + 2·21-s − 25-s + 4·27-s + 8·29-s + 2·35-s − 2·37-s − 4·39-s + 2·41-s + 4·43-s − 2·45-s − 47-s + 49-s − 12·51-s − 6·53-s + 2·57-s + 14·59-s − 6·61-s − 63-s − 4·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 1.03·15-s + 1.45·17-s − 0.229·19-s + 0.436·21-s − 1/5·25-s + 0.769·27-s + 1.48·29-s + 0.338·35-s − 0.328·37-s − 0.640·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s − 0.145·47-s + 1/7·49-s − 1.68·51-s − 0.824·53-s + 0.264·57-s + 1.82·59-s − 0.768·61-s − 0.125·63-s − 0.496·65-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 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−13.97704811659913, −13.52908241774755, −12.72380856632553, −12.42419147220131, −11.99227569473500, −11.64575169866439, −11.13720472561894, −10.56171083529461, −10.32335131176209, −9.618952267991816, −9.108495446077465, −8.305528685436249, −8.053909265587366, −7.495260342648930, −6.728886178342144, −6.459331290176532, −5.842320318100381, −5.340339539698338, −4.858523530734002, −4.142737444151854, −3.651130647195894, −3.089922677536598, −2.366680154717100, −1.241794122781707, −0.7820634336751139, 0, 0.7820634336751139, 1.241794122781707, 2.366680154717100, 3.089922677536598, 3.651130647195894, 4.142737444151854, 4.858523530734002, 5.340339539698338, 5.842320318100381, 6.459331290176532, 6.728886178342144, 7.495260342648930, 8.053909265587366, 8.305528685436249, 9.108495446077465, 9.618952267991816, 10.32335131176209, 10.56171083529461, 11.13720472561894, 11.64575169866439, 11.99227569473500, 12.42419147220131, 12.72380856632553, 13.52908241774755, 13.97704811659913

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