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Properties

Label 2-46200-1.1-c1-0-27
Degree $2$
Conductor $46200$
Sign $1$
Analytic cond. $368.908$
Root an. cond. $19.2070$
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-s − 7-s + 9-s − 11-s − 2·13-s + 2·17-s − 21-s + 4·23-s + 27-s + 6·29-s + 4·31-s − 33-s + 10·37-s − 2·39-s − 2·41-s + 4·43-s − 8·47-s + 49-s + 2·51-s − 2·53-s + 4·59-s − 10·61-s − 63-s − 4·67-s + 4·69-s + 8·71-s + 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.485·17-s − 0.218·21-s + 0.834·23-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.174·33-s + 1.64·37-s − 0.320·39-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.274·53-s + 0.520·59-s − 1.28·61-s − 0.125·63-s − 0.488·67-s + 0.481·69-s + 0.949·71-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46200\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(368.908\)
Root analytic conductor: \(19.2070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 46200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.823401621\)
\(L(\frac12)\) \(\approx\) \(2.823401621\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.66870302484408, −14.04860216277965, −13.68352021048932, −13.03177331874697, −12.61479359915981, −12.21673163075733, −11.44617684573038, −11.04486336192968, −10.27189165541948, −9.821962921673194, −9.547396129432835, −8.712334676053925, −8.371914702447093, −7.655781903334491, −7.315249437732451, −6.544536451571375, −6.130303683250780, −5.313188347898536, −4.704144509256794, −4.236630610846195, −3.306218987678486, −2.893993747287264, −2.338009208235197, −1.369180703581337, −0.6012024922825077, 0.6012024922825077, 1.369180703581337, 2.338009208235197, 2.893993747287264, 3.306218987678486, 4.236630610846195, 4.704144509256794, 5.313188347898536, 6.130303683250780, 6.544536451571375, 7.315249437732451, 7.655781903334491, 8.371914702447093, 8.712334676053925, 9.547396129432835, 9.821962921673194, 10.27189165541948, 11.04486336192968, 11.44617684573038, 12.21673163075733, 12.61479359915981, 13.03177331874697, 13.68352021048932, 14.04860216277965, 14.66870302484408

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