Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11 \cdot 19^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 3·5-s + 7-s + 6·10-s + 11-s − 4·13-s − 2·14-s − 4·16-s − 6·20-s − 2·22-s − 2·23-s + 4·25-s + 8·26-s + 2·28-s − 8·29-s − 5·31-s + 8·32-s − 3·35-s + 8·37-s + 41-s + 2·44-s + 4·46-s − 9·47-s + 49-s − 8·50-s − 8·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.34·5-s + 0.377·7-s + 1.89·10-s + 0.301·11-s − 1.10·13-s − 0.534·14-s − 16-s − 1.34·20-s − 0.426·22-s − 0.417·23-s + 4/5·25-s + 1.56·26-s + 0.377·28-s − 1.48·29-s − 0.898·31-s + 1.41·32-s − 0.507·35-s + 1.31·37-s + 0.156·41-s + 0.301·44-s + 0.589·46-s − 1.31·47-s + 1/7·49-s − 1.13·50-s − 1.10·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(250173\)    =    \(3^{2} \cdot 7 \cdot 11 \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{250173} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 250173,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
19 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−13.03192066220004, −12.47410178442069, −11.91031476953551, −11.51982072732814, −11.17737225405162, −10.91189670171413, −10.09245820945242, −9.794123577692051, −9.414800033477080, −8.784522815737501, −8.402097044917490, −7.972245818099821, −7.475039231465836, −7.290802769134794, −6.826440781737129, −6.055538556031689, −5.414763764910413, −4.834842277230634, −4.211252874085438, −3.968453037674966, −3.200547214476921, −2.494253019043569, −1.913703200021452, −1.319437187354474, −0.4989768599621284, 0, 0.4989768599621284, 1.319437187354474, 1.913703200021452, 2.494253019043569, 3.200547214476921, 3.968453037674966, 4.211252874085438, 4.834842277230634, 5.414763764910413, 6.055538556031689, 6.826440781737129, 7.290802769134794, 7.475039231465836, 7.972245818099821, 8.402097044917490, 8.784522815737501, 9.414800033477080, 9.794123577692051, 10.09245820945242, 10.91189670171413, 11.17737225405162, 11.51982072732814, 11.91031476953551, 12.47410178442069, 13.03192066220004

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