Properties

Degree $2$
Conductor $55470$
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 − 5-s − 6-s − 4·7-s + 8-s + 9-s − 10-s + 4·11-s − 12-s + 4·13-s − 4·14-s + 15-s + 16-s + 18-s − 20-s + 4·21-s + 4·22-s − 4·23-s − 24-s + 25-s + 4·26-s − 27-s − 4·28-s + 6·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 1.10·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.223·20-s + 0.872·21-s + 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55470\)    =    \(2 \cdot 3 \cdot 5 \cdot 43^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{55470} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 55470,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 + T \)
43 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.47263318989697, −14.16547783857115, −13.61948018154655, −12.91033459115711, −12.72590598888538, −12.21427031881439, −11.54340373861888, −11.35238973529811, −10.68355310008262, −10.00367831553033, −9.692132949339560, −8.971496994119880, −8.454697464596012, −7.731963099490854, −6.994090118106385, −6.570374919294184, −6.187695426093245, −5.821720793043551, −4.961009014017238, −4.259650594455860, −3.773928794601661, −3.396458186389253, −2.689172111636483, −1.664139852648633, −0.9506971920204913, 0, 0.9506971920204913, 1.664139852648633, 2.689172111636483, 3.396458186389253, 3.773928794601661, 4.259650594455860, 4.961009014017238, 5.821720793043551, 6.187695426093245, 6.570374919294184, 6.994090118106385, 7.731963099490854, 8.454697464596012, 8.971496994119880, 9.692132949339560, 10.00367831553033, 10.68355310008262, 11.35238973529811, 11.54340373861888, 12.21427031881439, 12.72590598888538, 12.91033459115711, 13.61948018154655, 14.16547783857115, 14.47263318989697

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