Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 6·11-s + 13-s + 16-s + 3·17-s − 2·19-s + 6·22-s − 26-s − 6·29-s + 4·31-s − 32-s − 3·34-s + 7·37-s + 2·38-s + 43-s − 6·44-s − 3·47-s + 52-s + 6·58-s − 6·59-s − 8·61-s − 4·62-s + 64-s − 14·67-s + 3·68-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 0.277·13-s + 1/4·16-s + 0.727·17-s − 0.458·19-s + 1.27·22-s − 0.196·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s + 1.15·37-s + 0.324·38-s + 0.152·43-s − 0.904·44-s − 0.437·47-s + 0.138·52-s + 0.787·58-s − 0.781·59-s − 1.02·61-s − 0.508·62-s + 1/8·64-s − 1.71·67-s + 0.363·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(286650\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{286650} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 286650,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4528361624$
$L(\frac12)$  $\approx$  $0.4528361624$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−12.86245939797886, −12.20207245103713, −11.78497345938778, −11.18260924557661, −10.77306052928358, −10.47984262588858, −10.00820439046329, −9.461365791700782, −9.163241432846129, −8.405411336728777, −8.045081173822213, −7.755169566508523, −7.309873321653619, −6.681562422327613, −6.106782368398046, −5.646939379279990, −5.275605874634425, −4.569206076632948, −4.104886814066737, −3.253094417791877, −2.874868445894739, −2.384989281253457, −1.691474395856619, −1.089633830226674, −0.2137988826278011, 0.2137988826278011, 1.089633830226674, 1.691474395856619, 2.384989281253457, 2.874868445894739, 3.253094417791877, 4.104886814066737, 4.569206076632948, 5.275605874634425, 5.646939379279990, 6.106782368398046, 6.681562422327613, 7.309873321653619, 7.755169566508523, 8.045081173822213, 8.405411336728777, 9.163241432846129, 9.461365791700782, 10.00820439046329, 10.47984262588858, 10.77306052928358, 11.18260924557661, 11.78497345938778, 12.20207245103713, 12.86245939797886

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