Properties

Label 2-47096-1.1-c1-0-12
Degree $2$
Conductor $47096$
Sign $1$
Analytic cond. $376.063$
Root an. cond. $19.3923$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 3·5-s + 7-s + 6·9-s + 3·11-s + 5·13-s + 9·15-s − 2·17-s + 4·19-s + 3·21-s + 4·23-s + 4·25-s + 9·27-s − 7·31-s + 9·33-s + 3·35-s + 2·37-s + 15·39-s + 8·41-s − 43-s + 18·45-s − 3·47-s + 49-s − 6·51-s + 9·53-s + 9·55-s + 12·57-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.34·5-s + 0.377·7-s + 2·9-s + 0.904·11-s + 1.38·13-s + 2.32·15-s − 0.485·17-s + 0.917·19-s + 0.654·21-s + 0.834·23-s + 4/5·25-s + 1.73·27-s − 1.25·31-s + 1.56·33-s + 0.507·35-s + 0.328·37-s + 2.40·39-s + 1.24·41-s − 0.152·43-s + 2.68·45-s − 0.437·47-s + 1/7·49-s − 0.840·51-s + 1.23·53-s + 1.21·55-s + 1.58·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47096\)    =    \(2^{3} \cdot 7 \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(376.063\)
Root analytic conductor: \(19.3923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{47096} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 47096,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.612830760\)
\(L(\frac12)\) \(\approx\) \(9.612830760\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
29 \( 1 \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−14.61408330207454, −14.01382962097981, −13.66137865132784, −13.18540874584272, −13.01130453257555, −12.09147898712303, −11.46255811674553, −10.73963707497550, −10.43077797374208, −9.495193791502353, −9.297790626305439, −8.960942669896910, −8.487655909299702, −7.726989722292273, −7.289186458955929, −6.607151830770315, −6.009602252833484, −5.489927252761990, −4.576439973738189, −4.038290443091339, −3.346868666423884, −2.876494922824304, −2.096355218772598, −1.524565562763103, −1.130986965877076, 1.130986965877076, 1.524565562763103, 2.096355218772598, 2.876494922824304, 3.346868666423884, 4.038290443091339, 4.576439973738189, 5.489927252761990, 6.009602252833484, 6.607151830770315, 7.289186458955929, 7.726989722292273, 8.487655909299702, 8.960942669896910, 9.297790626305439, 9.495193791502353, 10.43077797374208, 10.73963707497550, 11.46255811674553, 12.09147898712303, 13.01130453257555, 13.18540874584272, 13.66137865132784, 14.01382962097981, 14.61408330207454

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