Properties

Label 2-47096-1.1-c1-0-1
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  + 2·3-s − 2·5-s + 7-s + 9-s + 2·13-s − 4·15-s − 6·19-s + 2·21-s − 8·23-s − 25-s − 4·27-s − 6·31-s − 2·35-s − 10·37-s + 4·39-s + 12·41-s − 4·43-s − 2·45-s − 6·47-s + 49-s − 10·53-s − 12·57-s − 8·59-s + 12·61-s + 63-s − 4·65-s − 12·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.03·15-s − 1.37·19-s + 0.436·21-s − 1.66·23-s − 1/5·25-s − 0.769·27-s − 1.07·31-s − 0.338·35-s − 1.64·37-s + 0.640·39-s + 1.87·41-s − 0.609·43-s − 0.298·45-s − 0.875·47-s + 1/7·49-s − 1.37·53-s − 1.58·57-s − 1.04·59-s + 1.53·61-s + 0.125·63-s − 0.496·65-s − 1.46·67-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

\(L(1)\) \(\approx\) \(1.470828197\)
\(L(\frac12)\) \(\approx\) \(1.470828197\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−14.67962027179635, −14.06891016537994, −13.75303319089545, −13.12949726877938, −12.49644555153334, −12.13086894686499, −11.47564953151831, −10.87677098489618, −10.63132447372713, −9.667215083669694, −9.339046675639707, −8.576928366697067, −8.278398989871942, −7.886388279337553, −7.415063089597975, −6.606945201034037, −6.062591713779548, −5.360130652839226, −4.517186224810336, −3.935634744718093, −3.645639857303035, −2.942364597361550, −1.971029224753894, −1.810988999491984, −0.3734770943655791, 0.3734770943655791, 1.810988999491984, 1.971029224753894, 2.942364597361550, 3.645639857303035, 3.935634744718093, 4.517186224810336, 5.360130652839226, 6.062591713779548, 6.606945201034037, 7.415063089597975, 7.886388279337553, 8.278398989871942, 8.576928366697067, 9.339046675639707, 9.667215083669694, 10.63132447372713, 10.87677098489618, 11.47564953151831, 12.13086894686499, 12.49644555153334, 13.12949726877938, 13.75303319089545, 14.06891016537994, 14.67962027179635

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