Properties

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

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s − 2·11-s + 13-s − 14-s + 16-s + 3·17-s − 2·22-s − 23-s + 26-s − 28-s + 5·29-s + 7·31-s + 32-s + 3·34-s + 2·37-s − 7·41-s + 11·43-s − 2·44-s − 46-s + 8·47-s + 49-s + 52-s − 53-s − 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.603·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.426·22-s − 0.208·23-s + 0.196·26-s − 0.188·28-s + 0.928·29-s + 1.25·31-s + 0.176·32-s + 0.514·34-s + 0.328·37-s − 1.09·41-s + 1.67·43-s − 0.301·44-s − 0.147·46-s + 1.16·47-s + 1/7·49-s + 0.138·52-s − 0.137·53-s − 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.868416307$
$L(\frac12)$  $\approx$  $2.868416307$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−8.529164641594109163133395432045, −7.87684307158352638620482634323, −7.08385989826608745876101870372, −6.28226116608042795172243934117, −5.64696771527081683327099375104, −4.82521297061664361880487803844, −3.99423435966835080062884690070, −3.09739555959104785121840729688, −2.34801417757247614772000096677, −0.937153127736983851226818001742, 0.937153127736983851226818001742, 2.34801417757247614772000096677, 3.09739555959104785121840729688, 3.99423435966835080062884690070, 4.82521297061664361880487803844, 5.64696771527081683327099375104, 6.28226116608042795172243934117, 7.08385989826608745876101870372, 7.87684307158352638620482634323, 8.529164641594109163133395432045

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