Properties

Label 2-47190-1.1-c1-0-35
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 + 2·17-s − 18-s + 20-s − 2·21-s − 23-s + 24-s + 25-s + 26-s − 27-s + 2·28-s − 4·29-s + 30-s − 3·31-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 + 0.485·17-s − 0.235·18-s + 0.223·20-s − 0.436·21-s − 0.208·23-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 − 0.538·31-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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.74044276888013, −14.54308930576118, −13.85597167646274, −13.21474582802509, −12.75269039142000, −11.99983295551985, −11.81623015144634, −11.00441766995431, −10.82848444717372, −10.15627935693321, −9.567199990467699, −9.320985800409774, −8.384602389795111, −8.094058143260447, −7.460476830208318, −6.854667272546165, −6.384105629746123, −5.640272599393285, −5.208073863388251, −4.681162557404195, −3.774240269819915, −3.148132157644026, −2.118985389833654, −1.747281284935443, −0.9414820196658477, 0, 0.9414820196658477, 1.747281284935443, 2.118985389833654, 3.148132157644026, 3.774240269819915, 4.681162557404195, 5.208073863388251, 5.640272599393285, 6.384105629746123, 6.854667272546165, 7.460476830208318, 8.094058143260447, 8.384602389795111, 9.320985800409774, 9.567199990467699, 10.15627935693321, 10.82848444717372, 11.00441766995431, 11.81623015144634, 11.99983295551985, 12.75269039142000, 13.21474582802509, 13.85597167646274, 14.54308930576118, 14.74044276888013

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