Properties

Label 2-47190-1.1-c1-0-18
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 − 2·7-s − 8-s + 9-s + 10-s − 12-s + 13-s + 2·14-s + 15-s + 16-s + 4·17-s − 18-s + 2·19-s − 20-s + 2·21-s − 6·23-s + 24-s + 25-s − 26-s − 27-s − 2·28-s + 4·29-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 − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s + 0.534·14-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 + 0.436·21-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + 0.742·29-s − 0.182·30-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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.81199506002496, −14.55036004735304, −13.68658754307689, −13.33810012937449, −12.43814342905393, −12.30041315392268, −11.67898559097719, −11.32225213081628, −10.52465321561459, −10.07891461592279, −9.892465834565188, −9.063556733228209, −8.548913063335464, −8.008541786662589, −7.374449381684429, −6.926114828039224, −6.391527389700145, −5.658007569329412, −5.382894537182903, −4.401394187804597, −3.697700006651786, −3.277448814034030, −2.410303851634197, −1.553095447870013, −0.7741015905013478, 0, 0.7741015905013478, 1.553095447870013, 2.410303851634197, 3.277448814034030, 3.697700006651786, 4.401394187804597, 5.382894537182903, 5.658007569329412, 6.391527389700145, 6.926114828039224, 7.374449381684429, 8.008541786662589, 8.548913063335464, 9.063556733228209, 9.892465834565188, 10.07891461592279, 10.52465321561459, 11.32225213081628, 11.67898559097719, 12.30041315392268, 12.43814342905393, 13.33810012937449, 13.68658754307689, 14.55036004735304, 14.81199506002496

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