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 + 2.68·3-s + 4-s + 5-s − 2.68·6-s + 4.03·7-s − 8-s + 4.21·9-s − 10-s + 11-s + 2.68·12-s + 4.50·13-s − 4.03·14-s + 2.68·15-s + 16-s − 5.71·17-s − 4.21·18-s + 0.726·19-s + 20-s + 10.8·21-s − 22-s − 3.76·23-s − 2.68·24-s + 25-s − 4.50·26-s + 3.25·27-s + 4.03·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.55·3-s + 0.5·4-s + 0.447·5-s − 1.09·6-s + 1.52·7-s − 0.353·8-s + 1.40·9-s − 0.316·10-s + 0.301·11-s + 0.775·12-s + 1.25·13-s − 1.07·14-s + 0.693·15-s + 0.250·16-s − 1.38·17-s − 0.992·18-s + 0.166·19-s + 0.223·20-s + 2.36·21-s − 0.213·22-s − 0.784·23-s − 0.548·24-s + 0.200·25-s − 0.883·26-s + 0.626·27-s + 0.762·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.577119304$
$L(\frac12)$  $\approx$  $3.577119304$
$L(\frac{3}{2})$   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 \( 1 - 2.68T + 3T^{2} \)
7 \( 1 - 4.03T + 7T^{2} \)
13 \( 1 - 4.50T + 13T^{2} \)
17 \( 1 + 5.71T + 17T^{2} \)
19 \( 1 - 0.726T + 19T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 - 0.439T + 29T^{2} \)
31 \( 1 + 0.158T + 31T^{2} \)
37 \( 1 - 2.90T + 37T^{2} \)
41 \( 1 + 9.05T + 41T^{2} \)
47 \( 1 - 12.3T + 47T^{2} \)
53 \( 1 - 5.86T + 53T^{2} \)
59 \( 1 + 4.85T + 59T^{2} \)
61 \( 1 - 8.76T + 61T^{2} \)
67 \( 1 + 5.33T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 - 5.57T + 73T^{2} \)
79 \( 1 + 0.750T + 79T^{2} \)
83 \( 1 + 1.03T + 83T^{2} \)
89 \( 1 + 1.17T + 89T^{2} \)
97 \( 1 + 5.36T + 97T^{2} \)
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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

−8.436112645020118853778813689687, −7.916690254805017585886532483680, −7.13376476421605384027875261605, −6.34727662585317672112557454191, −5.37881563188727198393262214679, −4.30646354016045405894617915120, −3.70605633519931244929948049327, −2.51197565517043102503328696194, −1.96130245490876590485040592532, −1.20781715940695874317394570455, 1.20781715940695874317394570455, 1.96130245490876590485040592532, 2.51197565517043102503328696194, 3.70605633519931244929948049327, 4.30646354016045405894617915120, 5.37881563188727198393262214679, 6.34727662585317672112557454191, 7.13376476421605384027875261605, 7.916690254805017585886532483680, 8.436112645020118853778813689687

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