Properties

Degree 2
Conductor $ 2^{4} \cdot 7 \cdot 19 \cdot 47 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

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 − 2·17-s − 19-s − 2·21-s − 4·23-s − 25-s + 4·27-s − 8·29-s + 2·35-s − 10·37-s + 4·39-s − 2·41-s − 4·43-s + 2·45-s − 47-s + 49-s + 4·51-s − 6·53-s + 2·57-s − 10·59-s + 14·61-s + 63-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 − 0.485·17-s − 0.229·19-s − 0.436·21-s − 0.834·23-s − 1/5·25-s + 0.769·27-s − 1.48·29-s + 0.338·35-s − 1.64·37-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 + 0.560·51-s − 0.824·53-s + 0.264·57-s − 1.30·59-s + 1.79·61-s + 0.125·63-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  =  0
Selberg data  =  $(2,\ 100016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1075974915$
$L(\frac12)$  $\approx$  $0.1075974915$
$L(\frac{3}{2})$   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 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.66422753329309, −13.28439073836982, −12.71951696089543, −12.17193564025107, −11.82520841476332, −11.27681799519984, −10.88608131328861, −10.30757593161126, −9.970333837432331, −9.411283132961243, −8.815179182932072, −8.345084041137652, −7.593425764067315, −7.139644392605719, −6.448523329096879, −6.158768729213164, −5.420256921559849, −5.319096636740958, −4.617539600455230, −4.025201659060028, −3.265815275989085, −2.442210255531693, −1.825805328207019, −1.385146123144242, −0.1065124273002247, 0.1065124273002247, 1.385146123144242, 1.825805328207019, 2.442210255531693, 3.265815275989085, 4.025201659060028, 4.617539600455230, 5.319096636740958, 5.420256921559849, 6.158768729213164, 6.448523329096879, 7.139644392605719, 7.593425764067315, 8.345084041137652, 8.815179182932072, 9.411283132961243, 9.970333837432331, 10.30757593161126, 10.88608131328861, 11.27681799519984, 11.82520841476332, 12.17193564025107, 12.71951696089543, 13.28439073836982, 13.66422753329309

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