Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 11^{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 + 4·7-s − 3·8-s + 9-s − 12-s − 2·13-s + 4·14-s − 16-s − 2·17-s + 18-s + 4·21-s − 8·23-s − 3·24-s − 2·26-s + 27-s − 4·28-s + 6·29-s − 8·31-s + 5·32-s − 2·34-s − 36-s − 6·37-s − 2·39-s + 2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.872·21-s − 1.66·23-s − 0.612·24-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s − 1.43·31-s + 0.883·32-s − 0.342·34-s − 1/6·36-s − 0.986·37-s − 0.320·39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9075\)    =    \(3 \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9075} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 9075,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−17.32464789191630, −16.33447712783722, −15.75970315362046, −15.01452929763343, −14.66386064131894, −14.13800823758325, −13.82820913753735, −13.16393237821645, −12.34188465966345, −12.08112523016110, −11.30896931120900, −10.63851682563599, −9.899286204858027, −9.253841137940981, −8.589634209089395, −8.106825615260508, −7.567033943349277, −6.649894312335892, −5.818867299132806, −5.084512116506114, −4.586011638081625, −4.018980823513915, −3.193451769296080, −2.237188218905832, −1.527411183264230, 0, 1.527411183264230, 2.237188218905832, 3.193451769296080, 4.018980823513915, 4.586011638081625, 5.084512116506114, 5.818867299132806, 6.649894312335892, 7.567033943349277, 8.106825615260508, 8.589634209089395, 9.253841137940981, 9.899286204858027, 10.63851682563599, 11.30896931120900, 12.08112523016110, 12.34188465966345, 13.16393237821645, 13.82820913753735, 14.13800823758325, 14.66386064131894, 15.01452929763343, 15.75970315362046, 16.33447712783722, 17.32464789191630

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