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 − 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

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

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  =  $(2,\ 90,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.337595994$
$L(\frac12)$  $\approx$  $1.337595994$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
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

−19.86778692916680, −19.29091326364889, −18.02544692672595, −16.89821268427246, −15.93175318025324, −15.23541050144377, −13.78806788970128, −13.19720570989419, −12.30986793353698, −10.91913531404329, −9.880541566397139, −8.644229019078173, −6.731064707465642, −6.113979054225488, −4.326878799136916, −2.746287478759346, 2.746287478759346, 4.326878799136916, 6.113979054225488, 6.731064707465642, 8.644229019078173, 9.880541566397139, 10.91913531404329, 12.30986793353698, 13.19720570989419, 13.78806788970128, 15.23541050144377, 15.93175318025324, 16.89821268427246, 18.02544692672595, 19.29091326364889, 19.86778692916680

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