Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 4·11-s − 12-s − 13-s − 14-s + 15-s + 16-s + 2·17-s − 18-s + 19-s − 20-s − 21-s − 4·22-s + 4·23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.218·21-s − 0.852·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(51870\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{51870} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 51870,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 + T \)
19 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
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

−14.97923286115953, −14.21910200802438, −13.91236775679010, −12.95760364642788, −12.58820035162627, −11.98191167557162, −11.55465693365545, −11.23377575297898, −10.66700028844122, −10.10868410377237, −9.443739287164502, −9.117682947337804, −8.534759680275164, −7.798008806728391, −7.438581810190462, −6.847000343698017, −6.392184168212063, −5.613491482077174, −5.200187603769468, −4.346926048991616, −3.868647533770188, −3.151595470968984, −2.316590647151806, −1.425741372240115, −0.9811583287844444, 0, 0.9811583287844444, 1.425741372240115, 2.316590647151806, 3.151595470968984, 3.868647533770188, 4.346926048991616, 5.200187603769468, 5.613491482077174, 6.392184168212063, 6.847000343698017, 7.438581810190462, 7.798008806728391, 8.534759680275164, 9.117682947337804, 9.443739287164502, 10.10868410377237, 10.66700028844122, 11.23377575297898, 11.55465693365545, 11.98191167557162, 12.58820035162627, 12.95760364642788, 13.91236775679010, 14.21910200802438, 14.97923286115953

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