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 0

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 + 2·17-s − 18-s − 19-s − 20-s − 21-s + 6·23-s + 24-s + 25-s + 26-s − 27-s + 28-s − 6·29-s − 30-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 + 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.218·21-s + 1.25·23-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 + ⋯

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  =  0
Selberg data  =  $(2,\ 51870,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9616778313$
$L(\frac12)$  $\approx$  $0.9616778313$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−14.57137368680491, −14.21891101399247, −13.19249448366729, −12.84092972671878, −12.44624767577028, −11.65022690979601, −11.30907387727796, −11.03822978155822, −10.38725944154134, −9.788052059551715, −9.326367939635224, −8.805627787471036, −8.015113404449225, −7.764008793034319, −7.134504095218250, −6.646292922078989, −5.954158033272329, −5.417709026130159, −4.747066382410125, −4.235453715900604, −3.366725872792290, −2.803337802727986, −1.885117692405260, −1.221292529518161, −0.4357400790827405, 0.4357400790827405, 1.221292529518161, 1.885117692405260, 2.803337802727986, 3.366725872792290, 4.235453715900604, 4.747066382410125, 5.417709026130159, 5.954158033272329, 6.646292922078989, 7.134504095218250, 7.764008793034319, 8.015113404449225, 8.805627787471036, 9.326367939635224, 9.788052059551715, 10.38725944154134, 11.03822978155822, 11.30907387727796, 11.65022690979601, 12.44624767577028, 12.84092972671878, 13.19249448366729, 14.21891101399247, 14.57137368680491

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