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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s − 11-s + 12-s + 4·13-s + 2·14-s + 16-s + 6·17-s + 18-s + 4·19-s + 2·21-s − 22-s − 6·23-s + 24-s − 5·25-s + 4·26-s + 27-s + 2·28-s − 6·29-s − 8·31-s + 32-s − 33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.436·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.174·33-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 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.02889940689037, −13.77584213978275, −13.15295255634067, −12.81721560118132, −12.08546191268037, −11.77275787404324, −11.14780917994328, −10.87625094355762, −10.02563002538052, −9.752747409421075, −9.161779934290534, −8.337542331422547, −8.068505503264153, −7.525587277916313, −7.229152572617807, −6.252584330450435, −5.778414489637384, −5.424634261340309, −4.791811316210028, −4.000653093651710, −3.632039864502596, −3.199248978072920, −2.369906236722728, −1.509550597653004, −1.447237668144024, 0, 1.447237668144024, 1.509550597653004, 2.369906236722728, 3.199248978072920, 3.632039864502596, 4.000653093651710, 4.791811316210028, 5.424634261340309, 5.778414489637384, 6.252584330450435, 7.229152572617807, 7.525587277916313, 8.068505503264153, 8.337542331422547, 9.161779934290534, 9.752747409421075, 10.02563002538052, 10.87625094355762, 11.14780917994328, 11.77275787404324, 12.08546191268037, 12.81721560118132, 13.15295255634067, 13.77584213978275, 14.02889940689037

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