Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 $
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 + 2·7-s − 8-s − 10-s + 6·11-s − 4·13-s − 2·14-s + 16-s − 6·17-s − 4·19-s + 20-s − 6·22-s + 25-s + 4·26-s + 2·28-s − 6·29-s − 4·31-s − 32-s + 6·34-s + 2·35-s + 8·37-s + 4·38-s − 40-s + 8·43-s + 6·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s + 1.80·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s − 1.27·22-s + 1/5·25-s + 0.784·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.338·35-s + 1.31·37-s + 0.648·38-s − 0.158·40-s + 1.21·43-s + 0.904·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{90} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 90,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8199784570$
$L(\frac12)$  $\approx$  $0.8199784570$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.81937426910000, −19.07364118783349, −17.72802640536366, −17.34086711137603, −16.46206151977811, −14.90946690425789, −14.47336871923299, −12.97148624749449, −11.71789183657864, −10.91484441178598, −9.508626637108994, −8.808531037524113, −7.338595572196530, −6.210604122220950, −4.397457874928823, −1.984687420306805, 1.984687420306805, 4.397457874928823, 6.210604122220950, 7.338595572196530, 8.808531037524113, 9.508626637108994, 10.91484441178598, 11.71789183657864, 12.97148624749449, 14.47336871923299, 14.90946690425789, 16.46206151977811, 17.34086711137603, 17.72802640536366, 19.07364118783349, 19.81937426910000

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