Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 43^{2} $
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-s + 4-s + 4·5-s + 7-s + 8-s − 3·9-s + 4·10-s + 2·13-s + 14-s + 16-s + 6·17-s − 3·18-s − 4·19-s + 4·20-s + 11·25-s + 2·26-s + 28-s − 2·29-s + 2·31-s + 32-s + 6·34-s + 4·35-s − 3·36-s − 2·37-s − 4·38-s + 4·40-s − 2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.377·7-s + 0.353·8-s − 9-s + 1.26·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.917·19-s + 0.894·20-s + 11/5·25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s + 0.359·31-s + 0.176·32-s + 1.02·34-s + 0.676·35-s − 1/2·36-s − 0.328·37-s − 0.648·38-s + 0.632·40-s − 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(25886\)    =    \(2 \cdot 7 \cdot 43^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{25886} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 25886,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.067201582$
$L(\frac12)$  $\approx$  $6.067201582$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
43 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.93213760835141, −14.70779141364656, −14.12456270019432, −13.87312821326539, −13.21812991388925, −12.87524251047180, −12.11637998024009, −11.68602646023554, −10.85525664700532, −10.57144565345334, −9.957283227381822, −9.327439100888339, −8.784571428124875, −8.147386635670875, −7.534644100556365, −6.518516842338310, −6.207103568993711, −5.748162996917822, −5.129673207739386, −4.727528421275268, −3.535899874690122, −3.124795737381432, −2.207121280365740, −1.802625373872047, −0.8732888741475411, 0.8732888741475411, 1.802625373872047, 2.207121280365740, 3.124795737381432, 3.535899874690122, 4.727528421275268, 5.129673207739386, 5.748162996917822, 6.207103568993711, 6.518516842338310, 7.534644100556365, 8.147386635670875, 8.784571428124875, 9.327439100888339, 9.957283227381822, 10.57144565345334, 10.85525664700532, 11.68602646023554, 12.11637998024009, 12.87524251047180, 13.21812991388925, 13.87312821326539, 14.12456270019432, 14.70779141364656, 14.93213760835141

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