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 + 5-s + 7-s − 2·10-s − 11-s − 4·13-s − 2·14-s − 4·16-s − 6·17-s + 2·20-s + 2·22-s − 4·23-s − 4·25-s + 8·26-s + 2·28-s − 6·29-s − 31-s + 8·32-s + 12·34-s + 35-s − 2·37-s − 7·41-s − 10·43-s − 2·44-s + 8·46-s + 3·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.447·5-s + 0.377·7-s − 0.632·10-s − 0.301·11-s − 1.10·13-s − 0.534·14-s − 16-s − 1.45·17-s + 0.447·20-s + 0.426·22-s − 0.834·23-s − 4/5·25-s + 1.56·26-s + 0.377·28-s − 1.11·29-s − 0.179·31-s + 1.41·32-s + 2.05·34-s + 0.169·35-s − 0.328·37-s − 1.09·41-s − 1.52·43-s − 0.301·44-s + 1.17·46-s + 0.437·47-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 \( 1 - T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.25763392453295, −12.41423649280818, −12.09525905250254, −11.49649433476598, −11.10338658990757, −10.63900701689606, −10.20046347165225, −9.778328669444135, −9.314528707388651, −9.065057506666156, −8.372609185946495, −8.002629070060971, −7.643351735948142, −7.019120330310124, −6.685268416109710, −6.106069291760684, −5.415144201446785, −4.922809412132314, −4.472965081864065, −3.830959242065589, −3.069705867363797, −2.266078803066176, −1.911783644990202, −1.649109113430564, −0.4883137646203034, 0, 0.4883137646203034, 1.649109113430564, 1.911783644990202, 2.266078803066176, 3.069705867363797, 3.830959242065589, 4.472965081864065, 4.922809412132314, 5.415144201446785, 6.106069291760684, 6.685268416109710, 7.019120330310124, 7.643351735948142, 8.002629070060971, 8.372609185946495, 9.065057506666156, 9.314528707388651, 9.778328669444135, 10.20046347165225, 10.63900701689606, 11.10338658990757, 11.49649433476598, 12.09525905250254, 12.41423649280818, 13.25763392453295

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