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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 12-s + 2·13-s + 16-s − 6·17-s + 18-s − 8·19-s + 24-s + 2·26-s + 27-s + 6·29-s + 4·31-s + 32-s − 6·34-s + 36-s + 10·37-s − 8·38-s + 2·39-s + 6·41-s + 4·43-s + 48-s − 6·51-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 + 0.288·12-s + 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 1.83·19-s + 0.204·24-s + 0.392·26-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s + 1.64·37-s − 1.29·38-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.144·48-s − 0.840·51-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  :  $\chi_{7350} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7350,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.100959980\)
\(L(\frac12)\)  \(\approx\)  \(4.100959980\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
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

−17.14320717521769, −16.20020628732429, −15.92824100574718, −15.14771023899143, −14.82323573594306, −14.17999076499264, −13.45948079565668, −13.11203264491069, −12.59853131814703, −11.80990941397453, −11.06848716841951, −10.69271035402588, −9.909563063375618, −9.104005631691935, −8.444999569777664, −8.050439257637999, −6.988198267913483, −6.492916706602515, −5.948734138459188, −4.839266427707192, −4.246937520143006, −3.765338462314915, −2.454346301349809, −2.337772510540009, −0.8898495276700655, 0.8898495276700655, 2.337772510540009, 2.454346301349809, 3.765338462314915, 4.246937520143006, 4.839266427707192, 5.948734138459188, 6.492916706602515, 6.988198267913483, 8.050439257637999, 8.444999569777664, 9.104005631691935, 9.909563063375618, 10.69271035402588, 11.06848716841951, 11.80990941397453, 12.59853131814703, 13.11203264491069, 13.45948079565668, 14.17999076499264, 14.82323573594306, 15.14771023899143, 15.92824100574718, 16.20020628732429, 17.14320717521769

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