Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 11 \cdot 43 $
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-s − 1.40·3-s + 4-s + 5-s + 1.40·6-s + 3.56·7-s − 8-s − 1.03·9-s − 10-s + 11-s − 1.40·12-s − 2.43·13-s − 3.56·14-s − 1.40·15-s + 16-s + 0.571·17-s + 1.03·18-s + 3.41·19-s + 20-s − 4.99·21-s − 22-s − 4.43·23-s + 1.40·24-s + 25-s + 2.43·26-s + 5.65·27-s + 3.56·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.809·3-s + 0.5·4-s + 0.447·5-s + 0.572·6-s + 1.34·7-s − 0.353·8-s − 0.344·9-s − 0.316·10-s + 0.301·11-s − 0.404·12-s − 0.676·13-s − 0.952·14-s − 0.362·15-s + 0.250·16-s + 0.138·17-s + 0.243·18-s + 0.783·19-s + 0.223·20-s − 1.09·21-s − 0.213·22-s − 0.925·23-s + 0.286·24-s + 0.200·25-s + 0.478·26-s + 1.08·27-s + 0.673·28-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  =  1
Selberg data  =  $(2,\ 4730,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + 1.40T + 3T^{2} \)
7 \( 1 - 3.56T + 7T^{2} \)
13 \( 1 + 2.43T + 13T^{2} \)
17 \( 1 - 0.571T + 17T^{2} \)
19 \( 1 - 3.41T + 19T^{2} \)
23 \( 1 + 4.43T + 23T^{2} \)
29 \( 1 + 1.21T + 29T^{2} \)
31 \( 1 + 9.06T + 31T^{2} \)
37 \( 1 - 1.75T + 37T^{2} \)
41 \( 1 + 2.15T + 41T^{2} \)
47 \( 1 + 8.49T + 47T^{2} \)
53 \( 1 + 1.12T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 + 7.31T + 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 - 8.64T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−7.922342006724470463995667665164, −7.33882153399832115523223369882, −6.49865490230782460162994177136, −5.65071089237398221610031123824, −5.24493725492157731070757081422, −4.39695751309021040971506197257, −3.16078258611842075473906114672, −2.04591937735311447931992583862, −1.33346366859150099764648378213, 0, 1.33346366859150099764648378213, 2.04591937735311447931992583862, 3.16078258611842075473906114672, 4.39695751309021040971506197257, 5.24493725492157731070757081422, 5.65071089237398221610031123824, 6.49865490230782460162994177136, 7.33882153399832115523223369882, 7.922342006724470463995667665164

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