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 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 6-s + 7-s + 3·8-s + 9-s − 12-s + 6·13-s − 14-s − 16-s − 18-s − 8·19-s + 21-s + 8·23-s + 3·24-s − 6·26-s + 27-s − 28-s + 2·29-s − 4·31-s − 5·32-s − 36-s − 2·37-s + 8·38-s + 6·39-s + 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.288·12-s + 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.235·18-s − 1.83·19-s + 0.218·21-s + 1.66·23-s + 0.612·24-s − 1.17·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.718·31-s − 0.883·32-s − 1/6·36-s − 0.328·37-s + 1.29·38-s + 0.960·39-s + 0.937·41-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  =  1
Selberg data  =  $(2,\ 151725,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.44505223576158, −13.11152832280526, −12.84245510252921, −12.30805397985599, −11.28256147895762, −11.10032224439820, −10.69931588847894, −10.23049271794393, −9.540404034473901, −9.121821258982383, −8.739627603078266, −8.251046110993632, −8.121105522293883, −7.380897526815315, −6.649270053343217, −6.503360509887642, −5.542206702465660, −5.122659897477216, −4.301545339838313, −4.163934043463852, −3.406584312719423, −2.861597840828437, −1.937304057501721, −1.498811411530825, −0.9022297106658361, 0, 0.9022297106658361, 1.498811411530825, 1.937304057501721, 2.861597840828437, 3.406584312719423, 4.163934043463852, 4.301545339838313, 5.122659897477216, 5.542206702465660, 6.503360509887642, 6.649270053343217, 7.380897526815315, 8.121105522293883, 8.251046110993632, 8.739627603078266, 9.121821258982383, 9.540404034473901, 10.23049271794393, 10.69931588847894, 11.10032224439820, 11.28256147895762, 12.30805397985599, 12.84245510252921, 13.11152832280526, 13.44505223576158

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