Properties

Label 2-55470-1.1-c1-0-22
Degree $2$
Conductor $55470$
Sign $-1$
Analytic cond. $442.930$
Root an. cond. $21.0459$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 2·7-s − 8-s + 9-s + 10-s − 11-s + 12-s + 4·13-s + 2·14-s − 15-s + 16-s + 17-s − 18-s + 5·19-s − 20-s − 2·21-s + 22-s − 4·23-s − 24-s + 25-s − 4·26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.14·19-s − 0.223·20-s − 0.436·21-s + 0.213·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.377·28-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: $-1$
Analytic conductor: \(442.930\)
Root analytic conductor: \(21.0459\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−14.69911363856733, −14.15616058092493, −13.66092188926665, −13.11038055955506, −12.63424391956441, −12.10670117372293, −11.51402882711722, −10.98014672874003, −10.52556109355233, −9.885472211285627, −9.421005160110048, −9.061210631912894, −8.355144008311095, −7.946691066976729, −7.503686816505738, −6.808458436595420, −6.433055542916403, −5.576605394943334, −5.217609064884577, −4.051530586427270, −3.632067023904768, −3.202828187558438, −2.424599507736841, −1.669633601327868, −0.9077276484859419, 0, 0.9077276484859419, 1.669633601327868, 2.424599507736841, 3.202828187558438, 3.632067023904768, 4.051530586427270, 5.217609064884577, 5.576605394943334, 6.433055542916403, 6.808458436595420, 7.503686816505738, 7.946691066976729, 8.355144008311095, 9.061210631912894, 9.421005160110048, 9.885472211285627, 10.52556109355233, 10.98014672874003, 11.51402882711722, 12.10670117372293, 12.63424391956441, 13.11038055955506, 13.66092188926665, 14.15616058092493, 14.69911363856733

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