Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17 $
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 + 13-s + 4·14-s + 16-s − 17-s − 4·19-s + 20-s + 25-s − 26-s − 4·28-s − 6·29-s − 4·31-s − 32-s + 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 + 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s − 0.196·26-s − 0.755·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.171·34-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

\[\begin{aligned} \Lambda(s)=\mathstrut & 19890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 19890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(19890\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{19890} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 19890,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5810341480$
$L(\frac12)$  $\approx$  $0.5810341480$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
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

−15.95570149265086, −15.03390064834915, −14.90796215110039, −13.95247115103013, −13.31127453604250, −12.97046950922592, −12.48741190884352, −11.76725829377487, −11.08749101604532, −10.56489351845798, −10.01879169621713, −9.502595143816399, −9.074085963441593, −8.494920670752513, −7.792093097185218, −6.922824691263799, −6.693399563877043, −5.965559643658305, −5.525259924923441, −4.475655961031694, −3.639141219323810, −3.100506385531134, −2.265557956408865, −1.559437372860942, −0.3415309846634656, 0.3415309846634656, 1.559437372860942, 2.265557956408865, 3.100506385531134, 3.639141219323810, 4.475655961031694, 5.525259924923441, 5.965559643658305, 6.693399563877043, 6.922824691263799, 7.792093097185218, 8.494920670752513, 9.074085963441593, 9.502595143816399, 10.01879169621713, 10.56489351845798, 11.08749101604532, 11.76725829377487, 12.48741190884352, 12.97046950922592, 13.31127453604250, 13.95247115103013, 14.90796215110039, 15.03390064834915, 15.95570149265086

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