Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23 $
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 + 5-s + 7-s − 8-s − 10-s + 2·13-s − 14-s + 16-s − 6·17-s − 4·19-s + 20-s + 23-s + 25-s − 2·26-s + 28-s − 6·29-s − 4·31-s − 32-s + 6·34-s + 35-s + 2·37-s + 4·38-s − 40-s + 6·41-s + 8·43-s − 46-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.208·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.169·35-s + 0.328·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s + 1.21·43-s − 0.147·46-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(14490\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{14490} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 14490,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;23\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 - T \)
good11 \( 1 + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−16.59562811619855, −15.78479388150913, −15.43973926130773, −14.64421069078389, −14.39528797565163, −13.45206668289667, −12.97952083150712, −12.62670409921712, −11.53869076975415, −11.17762539117993, −10.81414819301900, −10.08375853087786, −9.414770799118966, −8.897647772735755, −8.458278718265369, −7.756407918977336, −7.026421064435071, −6.500826264717945, −5.854413720116718, −5.173975268977952, −4.275305689513048, −3.682457348381860, −2.480864505261472, −2.078641504430146, −1.145730288354950, 0, 1.145730288354950, 2.078641504430146, 2.480864505261472, 3.682457348381860, 4.275305689513048, 5.173975268977952, 5.854413720116718, 6.500826264717945, 7.026421064435071, 7.756407918977336, 8.458278718265369, 8.897647772735755, 9.414770799118966, 10.08375853087786, 10.81414819301900, 11.17762539117993, 11.53869076975415, 12.62670409921712, 12.97952083150712, 13.45206668289667, 14.39528797565163, 14.64421069078389, 15.43973926130773, 15.78479388150913, 16.59562811619855

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