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 + 12-s + 13-s − 14-s − 15-s + 16-s − 6·17-s − 18-s + 19-s − 20-s + 21-s − 24-s + 25-s − 26-s + 27-s + 28-s + 6·29-s + 30-s − 4·31-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 + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.218·21-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.182·30-s − 0.718·31-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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 - T \)
19 \( 1 - T \)
good11 \( 1 + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
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

−14.71528882022883, −14.38249763290644, −13.74384194702688, −13.16880192780379, −12.76111731075319, −12.05226829781911, −11.55679213652881, −11.07846200264149, −10.62870373625871, −10.06317602892849, −9.390330712063779, −8.958279458951882, −8.459330876109583, −8.098189466434214, −7.374584476759068, −7.058254914088917, −6.337540734103240, −5.821749797974669, −4.811681741748454, −4.469159316751511, −3.710138374418765, −3.034667796850677, −2.395255210232274, −1.737515275693739, −0.9602794677125935, 0, 0.9602794677125935, 1.737515275693739, 2.395255210232274, 3.034667796850677, 3.710138374418765, 4.469159316751511, 4.811681741748454, 5.821749797974669, 6.337540734103240, 7.058254914088917, 7.374584476759068, 8.098189466434214, 8.459330876109583, 8.958279458951882, 9.390330712063779, 10.06317602892849, 10.62870373625871, 11.07846200264149, 11.55679213652881, 12.05226829781911, 12.76111731075319, 13.16880192780379, 13.74384194702688, 14.38249763290644, 14.71528882022883

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