Properties

Label 2-47190-1.1-c1-0-16
Degree $2$
Conductor $47190$
Sign $-1$
Analytic cond. $376.814$
Root an. cond. $19.4116$
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 − 4·7-s − 8-s + 9-s − 10-s − 12-s − 13-s + 4·14-s − 15-s + 16-s + 2·17-s − 18-s + 20-s + 4·21-s − 4·23-s + 24-s + 25-s + 26-s − 27-s − 4·28-s − 10·29-s + 30-s − 32-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 − 0.288·12-s − 0.277·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.223·20-s + 0.872·21-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.755·28-s − 1.85·29-s + 0.182·30-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47190\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(376.814\)
Root analytic conductor: \(19.4116\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{47190} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 47190,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.92179277166721, −14.44138512607509, −13.68965278063926, −13.12869593602404, −12.78601612963843, −12.31131468466164, −11.60622060644365, −11.28659577253237, −10.50243053783872, −10.02428905913417, −9.678645463953093, −9.385070916904966, −8.596864401320103, −7.994445515833397, −7.298190885364399, −6.856021680608059, −6.301752255553938, −5.827498574553483, −5.358555824297126, −4.508780015184209, −3.562412468350297, −3.291925033240161, −2.292246123568283, −1.752589190319034, −0.7032665511454389, 0, 0.7032665511454389, 1.752589190319034, 2.292246123568283, 3.291925033240161, 3.562412468350297, 4.508780015184209, 5.358555824297126, 5.827498574553483, 6.301752255553938, 6.856021680608059, 7.298190885364399, 7.994445515833397, 8.596864401320103, 9.385070916904966, 9.678645463953093, 10.02428905913417, 10.50243053783872, 11.28659577253237, 11.60622060644365, 12.31131468466164, 12.78601612963843, 13.12869593602404, 13.68965278063926, 14.44138512607509, 14.92179277166721

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