Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{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-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 12-s + 2·13-s − 14-s + 16-s + 2·17-s + 18-s − 4·19-s − 21-s + 4·23-s + 24-s + 2·26-s + 27-s − 28-s + 6·29-s + 32-s + 2·34-s + 36-s − 2·37-s − 4·38-s + 2·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.218·21-s + 0.834·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.176·32-s + 0.342·34-s + 1/6·36-s − 0.328·37-s − 0.648·38-s + 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(127050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{127050} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 127050,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.114058809$
$L(\frac12)$  $\approx$  $6.114058809$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.41590963256619, −13.12905939127860, −12.69616673197419, −12.12171023724820, −11.78226747909114, −11.04091154404915, −10.64726659577532, −10.21516524320007, −9.624726701618714, −9.104751122676236, −8.475100239112001, −8.233789172800863, −7.516775041739269, −6.949515436640149, −6.530782702193371, −6.093691441212463, −5.324740251156884, −4.958858213748347, −4.210701637877399, −3.750778790455792, −3.288383347225880, −2.604562682736523, −2.183145899488535, −1.309174940339453, −0.6592439205982636, 0.6592439205982636, 1.309174940339453, 2.183145899488535, 2.604562682736523, 3.288383347225880, 3.750778790455792, 4.210701637877399, 4.958858213748347, 5.324740251156884, 6.093691441212463, 6.530782702193371, 6.949515436640149, 7.516775041739269, 8.233789172800863, 8.475100239112001, 9.104751122676236, 9.624726701618714, 10.21516524320007, 10.64726659577532, 11.04091154404915, 11.78226747909114, 12.12171023724820, 12.69616673197419, 13.12905939127860, 13.41590963256619

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