Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 11 \cdot 419 $
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 − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s + 10-s + 11-s − 3·12-s + 14-s − 3·15-s + 16-s + 4·17-s + 6·18-s − 8·19-s + 20-s − 3·21-s + 22-s − 4·23-s − 3·24-s + 25-s − 9·27-s + 28-s + 2·29-s − 3·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s + 0.301·11-s − 0.866·12-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.970·17-s + 1.41·18-s − 1.83·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 1.73·27-s + 0.188·28-s + 0.371·29-s − 0.547·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(46090\)    =    \(2 \cdot 5 \cdot 11 \cdot 419\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{46090} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 46090,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.377708914\)
\(L(\frac12)\)  \(\approx\)  \(2.377708914\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;419\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11,\;419\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 - T \)
419 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.73318961060870, −14.09730702770670, −13.38532466235516, −13.05191073965689, −12.34793175188604, −12.10566515502967, −11.65700423521963, −10.99315071307599, −10.67640813463889, −10.11165114400790, −9.745635435876501, −8.812023311509465, −8.142708711131244, −7.501234179607942, −6.851512523660849, −6.301783376537587, −5.907722150528568, −5.595895881447233, −4.745657784546937, −4.397786329691160, −3.909093975155204, −2.815303193207094, −2.054698117673528, −1.319622509672178, −0.5789330595222311, 0.5789330595222311, 1.319622509672178, 2.054698117673528, 2.815303193207094, 3.909093975155204, 4.397786329691160, 4.745657784546937, 5.595895881447233, 5.907722150528568, 6.301783376537587, 6.851512523660849, 7.501234179607942, 8.142708711131244, 8.812023311509465, 9.745635435876501, 10.11165114400790, 10.67640813463889, 10.99315071307599, 11.65700423521963, 12.10566515502967, 12.34793175188604, 13.05191073965689, 13.38532466235516, 14.09730702770670, 14.73318961060870

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