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 + 2·5-s + 7-s − 4·10-s − 11-s + 7·13-s − 2·14-s − 4·16-s + 5·17-s + 4·20-s + 2·22-s + 4·23-s − 25-s − 14·26-s + 2·28-s − 2·29-s − 4·31-s + 8·32-s − 10·34-s + 2·35-s − 4·37-s − 4·41-s + 12·43-s − 2·44-s − 8·46-s − 2·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.894·5-s + 0.377·7-s − 1.26·10-s − 0.301·11-s + 1.94·13-s − 0.534·14-s − 16-s + 1.21·17-s + 0.894·20-s + 0.426·22-s + 0.834·23-s − 1/5·25-s − 2.74·26-s + 0.377·28-s − 0.371·29-s − 0.718·31-s + 1.41·32-s − 1.71·34-s + 0.338·35-s − 0.657·37-s − 0.624·41-s + 1.82·43-s − 0.301·44-s − 1.17·46-s − 0.291·47-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 17 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−13.17686737401051, −12.53979786862343, −12.16426919345332, −11.25632724429347, −11.06562546004488, −10.80552748828811, −10.20016199376015, −9.714274091219108, −9.452552366546950, −8.822680449209103, −8.526657422453423, −8.094629132429073, −7.563990216096163, −7.098474712950700, −6.556318044402150, −5.967020628958622, −5.512760269861471, −5.189377665062029, −4.247727540540084, −3.803052119677973, −3.103949589544272, −2.481557514502335, −1.733854678588555, −1.354920649509282, −0.9617933187137398, 0, 0.9617933187137398, 1.354920649509282, 1.733854678588555, 2.481557514502335, 3.103949589544272, 3.803052119677973, 4.247727540540084, 5.189377665062029, 5.512760269861471, 5.967020628958622, 6.556318044402150, 7.098474712950700, 7.563990216096163, 8.094629132429073, 8.526657422453423, 8.822680449209103, 9.452552366546950, 9.714274091219108, 10.20016199376015, 10.80552748828811, 11.06562546004488, 11.25632724429347, 12.16426919345332, 12.53979786862343, 13.17686737401051

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