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 − 1.88·3-s + 4-s + 5-s + 1.88·6-s − 3.45·7-s − 8-s + 0.571·9-s − 10-s + 11-s − 1.88·12-s − 1.07·13-s + 3.45·14-s − 1.88·15-s + 16-s + 4.59·17-s − 0.571·18-s − 6.77·19-s + 20-s + 6.53·21-s − 22-s + 8.70·23-s + 1.88·24-s + 25-s + 1.07·26-s + 4.58·27-s − 3.45·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.09·3-s + 0.5·4-s + 0.447·5-s + 0.771·6-s − 1.30·7-s − 0.353·8-s + 0.190·9-s − 0.316·10-s + 0.301·11-s − 0.545·12-s − 0.298·13-s + 0.923·14-s − 0.487·15-s + 0.250·16-s + 1.11·17-s − 0.134·18-s − 1.55·19-s + 0.223·20-s + 1.42·21-s − 0.213·22-s + 1.81·23-s + 0.385·24-s + 0.200·25-s + 0.211·26-s + 0.883·27-s − 0.653·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4745444669$
$L(\frac12)$  $\approx$  $0.4745444669$
$L(\frac{3}{2})$   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 \( 1 + 1.88T + 3T^{2} \)
7 \( 1 + 3.45T + 7T^{2} \)
13 \( 1 + 1.07T + 13T^{2} \)
17 \( 1 - 4.59T + 17T^{2} \)
19 \( 1 + 6.77T + 19T^{2} \)
23 \( 1 - 8.70T + 23T^{2} \)
29 \( 1 + 5.36T + 29T^{2} \)
31 \( 1 + 4.60T + 31T^{2} \)
37 \( 1 + 7.21T + 37T^{2} \)
41 \( 1 + 7.42T + 41T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 + 7.57T + 61T^{2} \)
67 \( 1 - 6.54T + 67T^{2} \)
71 \( 1 + 9.42T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + 17.5T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 - 2.04T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−8.529330270673319801089433283185, −7.22305518659260408539536448137, −6.92502976644476902386161979802, −6.09038530600493259050210667168, −5.68002712638149775615910596381, −4.83055347940226097487061392558, −3.58296639078610139929703141964, −2.84337334441430772334286771202, −1.64113162598623176045817451033, −0.44519608781075129718803084684, 0.44519608781075129718803084684, 1.64113162598623176045817451033, 2.84337334441430772334286771202, 3.58296639078610139929703141964, 4.83055347940226097487061392558, 5.68002712638149775615910596381, 6.09038530600493259050210667168, 6.92502976644476902386161979802, 7.22305518659260408539536448137, 8.529330270673319801089433283185

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