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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 3·5-s − 7-s + 11-s + 4·16-s − 2·17-s − 6·20-s + 4·23-s + 4·25-s + 2·28-s + 5·31-s − 3·35-s − 10·37-s + 5·41-s + 4·43-s − 2·44-s − 3·47-s + 49-s + 5·53-s + 3·55-s − 10·59-s − 5·61-s − 8·64-s + 10·67-s + 4·68-s + 15·71-s + 14·73-s + ⋯
L(s)  = 1  − 4-s + 1.34·5-s − 0.377·7-s + 0.301·11-s + 16-s − 0.485·17-s − 1.34·20-s + 0.834·23-s + 4/5·25-s + 0.377·28-s + 0.898·31-s − 0.507·35-s − 1.64·37-s + 0.780·41-s + 0.609·43-s − 0.301·44-s − 0.437·47-s + 1/7·49-s + 0.686·53-s + 0.404·55-s − 1.30·59-s − 0.640·61-s − 64-s + 1.22·67-s + 0.485·68-s + 1.78·71-s + 1.63·73-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  =  \(0\)
Selberg data  =  \((2,\ 250173,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.897552286\)
\(L(\frac12)\)  \(\approx\)  \(2.897552286\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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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

−12.91591241141738, −12.40953316983289, −12.25733210660333, −11.40035384686135, −10.82734732552241, −10.47281401532198, −9.986348907471993, −9.480207231976972, −9.174251481651412, −8.945894632274770, −8.186117569327384, −7.865560800028886, −7.021506340865609, −6.559201490675785, −6.248785538991981, −5.538209627985356, −5.231716746791032, −4.739361408653305, −4.152174712125772, −3.527494083488037, −3.053706589300308, −2.299548844299075, −1.841553497018769, −1.027666250283398, −0.5399997963191838, 0.5399997963191838, 1.027666250283398, 1.841553497018769, 2.299548844299075, 3.053706589300308, 3.527494083488037, 4.152174712125772, 4.739361408653305, 5.231716746791032, 5.538209627985356, 6.248785538991981, 6.559201490675785, 7.021506340865609, 7.865560800028886, 8.186117569327384, 8.945894632274770, 9.174251481651412, 9.480207231976972, 9.986348907471993, 10.47281401532198, 10.82734732552241, 11.40035384686135, 12.25733210660333, 12.40953316983289, 12.91591241141738

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