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 + 0.921·3-s + 4-s − 5-s + 0.921·6-s − 3.84·7-s + 8-s − 2.15·9-s − 10-s + 11-s + 0.921·12-s + 3.36·13-s − 3.84·14-s − 0.921·15-s + 16-s + 4.48·17-s − 2.15·18-s − 6.54·19-s − 20-s − 3.54·21-s + 22-s + 2.73·23-s + 0.921·24-s + 25-s + 3.36·26-s − 4.74·27-s − 3.84·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.532·3-s + 0.5·4-s − 0.447·5-s + 0.376·6-s − 1.45·7-s + 0.353·8-s − 0.716·9-s − 0.316·10-s + 0.301·11-s + 0.266·12-s + 0.932·13-s − 1.02·14-s − 0.237·15-s + 0.250·16-s + 1.08·17-s − 0.506·18-s − 1.50·19-s − 0.223·20-s − 0.772·21-s + 0.213·22-s + 0.571·23-s + 0.188·24-s + 0.200·25-s + 0.659·26-s − 0.913·27-s − 0.726·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.726452196$
$L(\frac12)$  $\approx$  $2.726452196$
$L(\frac{3}{2})$   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 \( 1 - 0.921T + 3T^{2} \)
7 \( 1 + 3.84T + 7T^{2} \)
13 \( 1 - 3.36T + 13T^{2} \)
17 \( 1 - 4.48T + 17T^{2} \)
19 \( 1 + 6.54T + 19T^{2} \)
23 \( 1 - 2.73T + 23T^{2} \)
29 \( 1 - 2.61T + 29T^{2} \)
31 \( 1 + 0.588T + 31T^{2} \)
37 \( 1 + 0.240T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
47 \( 1 - 9.27T + 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 + 9.61T + 61T^{2} \)
67 \( 1 + 8.11T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 - 5.98T + 79T^{2} \)
83 \( 1 - 8.56T + 83T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 + 8.94T + 97T^{2} \)
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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

−8.331311083860331222807306405996, −7.49423073815917276473559889617, −6.71021925854088288854341154459, −6.06510276384026228861638993670, −5.55888449043772419480066683671, −4.25373744405142039508612158823, −3.70474681778706273801281176176, −3.06506044231373806997051682747, −2.34521341721808445025243822474, −0.78166653688493349107480652763, 0.78166653688493349107480652763, 2.34521341721808445025243822474, 3.06506044231373806997051682747, 3.70474681778706273801281176176, 4.25373744405142039508612158823, 5.55888449043772419480066683671, 6.06510276384026228861638993670, 6.71021925854088288854341154459, 7.49423073815917276473559889617, 8.331311083860331222807306405996

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