Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
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 + 3-s + 4-s − 6-s − 8-s + 9-s + 2·11-s + 12-s − 6·13-s + 16-s − 2·17-s − 18-s − 2·22-s − 4·23-s − 24-s + 6·26-s + 27-s + 8·31-s − 32-s + 2·33-s + 2·34-s + 36-s + 2·37-s − 6·39-s − 2·41-s − 4·43-s + 2·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s − 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.426·22-s − 0.834·23-s − 0.204·24-s + 1.17·26-s + 0.192·27-s + 1.43·31-s − 0.176·32-s + 0.348·33-s + 0.342·34-s + 1/6·36-s + 0.328·37-s − 0.960·39-s − 0.312·41-s − 0.609·43-s + 0.301·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7350} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7350,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.504712335\)
\(L(\frac12)\)  \(\approx\)  \(1.504712335\)
\(L(\frac{3}{2})\)   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 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−7.82098831101325334089445349316, −7.46117968917097988560252453746, −6.65884956600536742374760867539, −6.07449955694113876093085627684, −4.95374750465246110684844185298, −4.34625181903750527013463544138, −3.36859006421653416417174670158, −2.48915679641844057992520515579, −1.90961960649814903225932183911, −0.65092090841378482411444159399, 0.65092090841378482411444159399, 1.90961960649814903225932183911, 2.48915679641844057992520515579, 3.36859006421653416417174670158, 4.34625181903750527013463544138, 4.95374750465246110684844185298, 6.07449955694113876093085627684, 6.65884956600536742374760867539, 7.46117968917097988560252453746, 7.82098831101325334089445349316

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