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 + 8·31-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 + 1.43·31-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 \( 1 - 8 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−14.99421579514503, −14.28091297966435, −13.80664447485091, −13.26404309652022, −12.52043039490253, −12.18869698762425, −11.56078476649479, −11.15844348386861, −10.49142561905427, −10.05111730520604, −9.453673853315489, −9.072996438053264, −8.312572310214611, −7.990884074010785, −7.571723250046816, −7.057244934408685, −6.274262688983821, −5.745655970639752, −5.030604263311790, −4.316958662472679, −3.684677641685987, −3.036932318322469, −2.533723359912700, −1.487183539555121, −1.138262467366271, 0, 1.138262467366271, 1.487183539555121, 2.533723359912700, 3.036932318322469, 3.684677641685987, 4.316958662472679, 5.030604263311790, 5.745655970639752, 6.274262688983821, 7.057244934408685, 7.571723250046816, 7.990884074010785, 8.312572310214611, 9.072996438053264, 9.453673853315489, 10.05111730520604, 10.49142561905427, 11.15844348386861, 11.56078476649479, 12.18869698762425, 12.52043039490253, 13.26404309652022, 13.80664447485091, 14.28091297966435, 14.99421579514503

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