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 + 3-s + 4-s + 5-s + 6-s + 5·7-s + 8-s − 2·9-s + 10-s + 11-s + 12-s + 4·13-s + 5·14-s + 15-s + 16-s + 17-s − 2·18-s + 7·19-s + 20-s + 5·21-s + 22-s + 8·23-s + 24-s + 25-s + 4·26-s − 5·27-s + 5·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.88·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 1.33·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.471·18-s + 1.60·19-s + 0.223·20-s + 1.09·21-s + 0.213·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.962·27-s + 0.944·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.590070692$
$L(\frac12)$  $\approx$  $5.590070692$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.293635854471329195384609932348, −7.53375844824489400308132087217, −6.99024279853805331316094490451, −5.70039217147278135701040872176, −5.41370680067114110948002765045, −4.72246126695946412008487609416, −3.58198890718384045082579882256, −3.11570918719347193540994950175, −1.81571818368577660326243258448, −1.39559021097440837064201338799, 1.39559021097440837064201338799, 1.81571818368577660326243258448, 3.11570918719347193540994950175, 3.58198890718384045082579882256, 4.72246126695946412008487609416, 5.41370680067114110948002765045, 5.70039217147278135701040872176, 6.99024279853805331316094490451, 7.53375844824489400308132087217, 8.293635854471329195384609932348

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