Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 37^{2} $
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 + 2·5-s − 6-s − 4·7-s + 8-s + 9-s + 2·10-s + 11-s − 12-s − 6·13-s − 4·14-s − 2·15-s + 16-s + 2·17-s + 18-s + 2·20-s + 4·21-s + 22-s + 8·23-s − 24-s − 25-s − 6·26-s − 27-s − 4·28-s − 10·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s − 1.66·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.447·20-s + 0.872·21-s + 0.213·22-s + 1.66·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.755·28-s − 1.85·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(90354\)    =    \(2 \cdot 3 \cdot 11 \cdot 37^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{90354} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 90354,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;37\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;37\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
11 \( 1 - T \)
37 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−14.13976532066287, −13.39795547656480, −12.95945849170021, −12.79012635809470, −12.40897435910629, −11.66373346679961, −11.26716237416365, −10.67707425053021, −10.02787199259262, −9.729322058351834, −9.322331721495451, −8.910046547992742, −7.708308794578952, −7.326445163895042, −6.871530121230711, −6.411657202609238, −5.814353709334440, −5.309292449076895, −5.109326010584727, −4.170126376930922, −3.573285119635715, −3.081782907533919, −2.323470619212207, −1.884740111971427, −0.8392275589318743, 0, 0.8392275589318743, 1.884740111971427, 2.323470619212207, 3.081782907533919, 3.573285119635715, 4.170126376930922, 5.109326010584727, 5.309292449076895, 5.814353709334440, 6.411657202609238, 6.871530121230711, 7.326445163895042, 7.708308794578952, 8.910046547992742, 9.322331721495451, 9.729322058351834, 10.02787199259262, 10.67707425053021, 11.26716237416365, 11.66373346679961, 12.40897435910629, 12.79012635809470, 12.95945849170021, 13.39795547656480, 14.13976532066287

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