Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2} $
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 − 7-s + 8-s + 10-s − 14-s + 16-s − 6·17-s + 4·19-s + 20-s + 25-s − 28-s − 6·29-s + 4·31-s + 32-s − 6·34-s − 35-s + 10·37-s + 4·38-s + 40-s + 6·41-s + 8·43-s + 49-s + 50-s + 6·53-s − 56-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 − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.223·20-s + 1/5·25-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 + 1.64·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s + 1.21·43-s + 1/7·49-s + 0.141·50-s + 0.824·53-s − 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(106470\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{106470} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 106470,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.347831362\)
\(L(\frac12)\)  \(\approx\)  \(4.347831362\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
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

−13.58470931309286, −13.21617866766242, −12.88273751394141, −12.35203578686250, −11.75836795574177, −11.21634075649748, −10.96273628720309, −10.34035814273864, −9.596913915408386, −9.448232697902461, −8.824417363545701, −8.157082967251539, −7.538160155276926, −7.114332783765932, −6.511742525081436, −6.036243909225910, −5.617418131851085, −5.002656337549484, −4.328969100437512, −4.004263062103637, −3.228480896420856, −2.525154633228429, −2.275667086472099, −1.319953246440618, −0.5810635866907767, 0.5810635866907767, 1.319953246440618, 2.275667086472099, 2.525154633228429, 3.228480896420856, 4.004263062103637, 4.328969100437512, 5.002656337549484, 5.617418131851085, 6.036243909225910, 6.511742525081436, 7.114332783765932, 7.538160155276926, 8.157082967251539, 8.824417363545701, 9.448232697902461, 9.596913915408386, 10.34035814273864, 10.96273628720309, 11.21634075649748, 11.75836795574177, 12.35203578686250, 12.88273751394141, 13.21617866766242, 13.58470931309286

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