Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19^{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 − 2·13-s − 14-s + 16-s + 6·17-s − 20-s + 25-s + 2·26-s + 28-s + 6·29-s + 4·31-s − 32-s − 6·34-s − 35-s + 10·37-s + 40-s − 6·41-s − 4·43-s + 49-s − 50-s − 2·52-s − 6·53-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 − 0.223·20-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 + 1.64·37-s + 0.158·40-s − 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.141·50-s − 0.277·52-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(227430\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{227430} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 227430,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.246822652\)
\(L(\frac12)\)  \(\approx\)  \(1.246822652\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;19\}$, 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 \( 1 - T \)
19 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
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

−12.70657316531154, −12.37264703363554, −11.96217435144244, −11.50667544650503, −11.15042154127463, −10.46814586429325, −10.13160659401885, −9.718904057090997, −9.255343026184418, −8.605844359108575, −8.145071508571578, −7.849016813387354, −7.417690154292063, −6.865890686357388, −6.261289725741893, −5.886954498235628, −5.101478919356744, −4.754565283335015, −4.164116497684384, −3.434445939036761, −2.888560058740842, −2.525572381573834, −1.487881788907228, −1.230397176926153, −0.3702598781077162, 0.3702598781077162, 1.230397176926153, 1.487881788907228, 2.525572381573834, 2.888560058740842, 3.434445939036761, 4.164116497684384, 4.754565283335015, 5.101478919356744, 5.886954498235628, 6.261289725741893, 6.865890686357388, 7.417690154292063, 7.849016813387354, 8.145071508571578, 8.605844359108575, 9.255343026184418, 9.718904057090997, 10.13160659401885, 10.46814586429325, 11.15042154127463, 11.50667544650503, 11.96217435144244, 12.37264703363554, 12.70657316531154

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