Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 1063 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 2·11-s + 12-s + 14-s − 15-s + 16-s − 3·17-s + 18-s − 8·19-s − 20-s + 21-s − 2·22-s − 4·23-s + 24-s − 4·25-s + 27-s + 28-s + 9·29-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 − 0.603·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.188·28-s + 1.67·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6378\)    =    \(2 \cdot 3 \cdot 1063\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6378} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 6378,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;1063\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;1063\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
1063 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.64441057203981, −17.00802262892521, −16.18333484674232, −15.63888583162084, −15.27397542670459, −14.67136499175648, −14.04629537240327, −13.48338631903846, −12.95770252901083, −12.30209090440613, −11.69438913598900, −11.07694308489920, −10.29799578699947, −9.928315265197049, −8.718157945878418, −8.227608008984284, −7.893004439582036, −6.677609248380151, −6.524745432918867, −5.374854909183303, −4.566639051633140, −4.164804470236657, −3.242444105113876, −2.401509237427546, −1.705466635262207, 0, 1.705466635262207, 2.401509237427546, 3.242444105113876, 4.164804470236657, 4.566639051633140, 5.374854909183303, 6.524745432918867, 6.677609248380151, 7.893004439582036, 8.227608008984284, 8.718157945878418, 9.928315265197049, 10.29799578699947, 11.07694308489920, 11.69438913598900, 12.30209090440613, 12.95770252901083, 13.48338631903846, 14.04629537240327, 14.67136499175648, 15.27397542670459, 15.63888583162084, 16.18333484674232, 17.00802262892521, 17.64441057203981

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