Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 $
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 + 4-s − 2·7-s − 8-s + 11-s + 4·13-s + 2·14-s + 16-s − 6·17-s − 4·19-s − 22-s + 6·23-s − 4·26-s − 2·28-s − 6·29-s + 8·31-s − 32-s + 6·34-s + 10·37-s + 4·38-s − 6·41-s − 8·43-s + 44-s − 6·46-s − 6·47-s − 3·49-s + 4·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.301·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.213·22-s + 1.25·23-s − 0.784·26-s − 0.377·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s + 1.64·37-s + 0.648·38-s − 0.937·41-s − 1.21·43-s + 0.150·44-s − 0.884·46-s − 0.875·47-s − 3/7·49-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4950\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4950} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4950,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−18.04464914005969, −17.49460184444710, −16.77480855218428, −16.44291333051628, −15.64776751239761, −15.19372978327370, −14.67322096280190, −13.47782107281249, −13.23884059721932, −12.66537855350138, −11.56847641783397, −11.25146445812561, −10.61162026354681, −9.810437946823134, −9.274626687170415, −8.564919537963217, −8.180244780572686, −7.015158370816194, −6.536951378468722, −6.098164154909148, −4.930163081259852, −4.041975704014233, −3.205870848224007, −2.306676276188877, −1.241375727875185, 0, 1.241375727875185, 2.306676276188877, 3.205870848224007, 4.041975704014233, 4.930163081259852, 6.098164154909148, 6.536951378468722, 7.015158370816194, 8.180244780572686, 8.564919537963217, 9.274626687170415, 9.810437946823134, 10.61162026354681, 11.25146445812561, 11.56847641783397, 12.66537855350138, 13.23884059721932, 13.47782107281249, 14.67322096280190, 15.19372978327370, 15.64776751239761, 16.44291333051628, 16.77480855218428, 17.49460184444710, 18.04464914005969

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