Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 23^{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 + 4-s − 5-s + 4·7-s + 8-s − 10-s + 2·13-s + 4·14-s + 16-s + 6·17-s + 4·19-s − 20-s + 25-s + 2·26-s + 4·28-s + 6·29-s + 8·31-s + 32-s + 6·34-s − 4·35-s − 2·37-s + 4·38-s − 40-s + 6·41-s + 4·43-s + 9·49-s + 50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.755·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s − 0.676·35-s − 0.328·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s + 0.609·43-s + 9/7·49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(47610\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{47610} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 47610,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(6.456434411\)
\(L(\frac12)\)  \(\approx\)  \(6.456434411\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
23 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.56562507638365, −14.11783443837333, −13.71264538609088, −13.13227769158495, −12.28370026801304, −12.02428439953040, −11.68699382876130, −10.97012637544470, −10.72453903162217, −9.968643852427164, −9.447020472108400, −8.483454844539392, −8.202287525112532, −7.701156031055550, −7.241268666852408, −6.444759953208820, −5.868186861375532, −5.175496384994778, −4.879872195131425, −4.193382393966589, −3.607924183484594, −2.936522099929823, −2.260409003977439, −1.199992779369645, −0.9938401776343899, 0.9938401776343899, 1.199992779369645, 2.260409003977439, 2.936522099929823, 3.607924183484594, 4.193382393966589, 4.879872195131425, 5.175496384994778, 5.868186861375532, 6.444759953208820, 7.241268666852408, 7.701156031055550, 8.202287525112532, 8.483454844539392, 9.447020472108400, 9.968643852427164, 10.72453903162217, 10.97012637544470, 11.68699382876130, 12.02428439953040, 12.28370026801304, 13.13227769158495, 13.71264538609088, 14.11783443837333, 14.56562507638365

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