Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 \cdot 17^{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  − 3-s − 2·4-s + 7-s + 9-s − 2·11-s + 2·12-s + 5·13-s + 4·16-s + 2·19-s − 21-s − 23-s − 27-s − 2·28-s − 8·29-s − 31-s + 2·33-s − 2·36-s − 3·37-s − 5·39-s + 7·41-s + 4·44-s + 47-s − 4·48-s + 49-s − 10·52-s + 8·53-s − 2·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.577·12-s + 1.38·13-s + 16-s + 0.458·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s − 0.377·28-s − 1.48·29-s − 0.179·31-s + 0.348·33-s − 1/3·36-s − 0.493·37-s − 0.800·39-s + 1.09·41-s + 0.603·44-s + 0.145·47-s − 0.577·48-s + 1/7·49-s − 1.38·52-s + 1.09·53-s − 0.264·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(151725\)    =    \(3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{151725} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 151725,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.311467994\)
\(L(\frac12)\)  \(\approx\)  \(1.311467994\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
17 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 13 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.27361884179692, −13.06966399178675, −12.27722463699945, −12.06682572261492, −11.28900701665432, −10.91824196769036, −10.61144063505013, −9.929274414219173, −9.532987330506472, −8.912653643324309, −8.605617761485701, −7.987445962744085, −7.544034689137062, −7.072930134832477, −6.195965279947584, −5.836497489821203, −5.376055548793203, −4.957381373153672, −4.248070437879983, −3.820277718237914, −3.376364839845953, −2.492254659692607, −1.683102192302730, −1.081129783599351, −0.4117448433699520, 0.4117448433699520, 1.081129783599351, 1.683102192302730, 2.492254659692607, 3.376364839845953, 3.820277718237914, 4.248070437879983, 4.957381373153672, 5.376055548793203, 5.836497489821203, 6.195965279947584, 7.072930134832477, 7.544034689137062, 7.987445962744085, 8.605617761485701, 8.912653643324309, 9.532987330506472, 9.929274414219173, 10.61144063505013, 10.91824196769036, 11.28900701665432, 12.06682572261492, 12.27722463699945, 13.06966399178675, 13.27361884179692

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