Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 11 \cdot 43 $
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 − 2·3-s + 4-s + 5-s + 2·6-s − 8-s + 9-s − 10-s + 11-s − 2·12-s + 2·13-s − 2·15-s + 16-s + 6·17-s − 18-s + 6·19-s + 20-s − 22-s + 6·23-s + 2·24-s + 25-s − 2·26-s + 4·27-s + 2·29-s + 2·30-s − 32-s − 2·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.577·12-s + 0.554·13-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 1.37·19-s + 0.223·20-s − 0.213·22-s + 1.25·23-s + 0.408·24-s + 1/5·25-s − 0.392·26-s + 0.769·27-s + 0.371·29-s + 0.365·30-s − 0.176·32-s − 0.348·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4730\)    =    \(2 \cdot 5 \cdot 11 \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4730} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.174774096$
$L(\frac12)$  $\approx$  $1.174774096$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−8.336939949401875147328489769576, −7.49147475343850383252662845938, −6.80266490482020843146257269043, −6.15879496806389957289656657750, −5.40155792313307253521379591553, −5.03149033176124286256076983222, −3.60562690516649849352622546792, −2.84884050107761652319790436586, −1.41100236300005680256219100842, −0.804039800970754814799351058596, 0.804039800970754814799351058596, 1.41100236300005680256219100842, 2.84884050107761652319790436586, 3.60562690516649849352622546792, 5.03149033176124286256076983222, 5.40155792313307253521379591553, 6.15879496806389957289656657750, 6.80266490482020843146257269043, 7.49147475343850383252662845938, 8.336939949401875147328489769576

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