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 − 2·11-s − 12-s − 6·13-s − 4·14-s − 15-s + 16-s − 4·17-s − 18-s + 2·19-s + 20-s − 4·21-s + 2·22-s − 4·23-s + 24-s + 25-s + 6·26-s − 27-s + 4·28-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 − 0.603·11-s − 0.288·12-s − 1.66·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s − 0.872·21-s + 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.755·28-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−14.74141002593637, −14.20071386929700, −13.70873153824259, −13.13093230070431, −12.35052208690938, −11.96714861113270, −11.64826208726496, −10.92832168367230, −10.59007040342061, −10.08082838740741, −9.569179695800139, −9.036763212615239, −8.269063396116345, −7.881158907258691, −7.499017639689858, −6.747361727437593, −6.333258916677593, −5.469037197594573, −5.011711540131577, −4.698608926087184, −3.929446786149432, −2.694209047742165, −2.332514863911745, −1.707023462692719, −0.8871759426454967, 0, 0.8871759426454967, 1.707023462692719, 2.332514863911745, 2.694209047742165, 3.929446786149432, 4.698608926087184, 5.011711540131577, 5.469037197594573, 6.333258916677593, 6.747361727437593, 7.499017639689858, 7.881158907258691, 8.269063396116345, 9.036763212615239, 9.569179695800139, 10.08082838740741, 10.59007040342061, 10.92832168367230, 11.64826208726496, 11.96714861113270, 12.35052208690938, 13.13093230070431, 13.70873153824259, 14.20071386929700, 14.74141002593637

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