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Properties

Label 2-46200-1.1-c1-0-14
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 − 8·17-s + 8·19-s − 21-s − 6·23-s + 27-s + 2·29-s + 4·31-s − 33-s + 10·41-s − 4·43-s + 4·47-s + 49-s − 8·51-s + 6·53-s + 8·57-s − 6·61-s − 63-s − 2·67-s − 6·69-s − 8·71-s + 2·73-s + 77-s − 12·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.94·17-s + 1.83·19-s − 0.218·21-s − 1.25·23-s + 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.174·33-s + 1.56·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s − 1.12·51-s + 0.824·53-s + 1.05·57-s − 0.768·61-s − 0.125·63-s − 0.244·67-s − 0.722·69-s − 0.949·71-s + 0.234·73-s + 0.113·77-s − 1.35·79-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

\(L(1)\) \(\approx\) \(2.236792167\)
\(L(\frac12)\) \(\approx\) \(2.236792167\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−14.57298686440132, −13.94118175909205, −13.63016165380054, −13.24830747817822, −12.64237604059996, −11.94810874435642, −11.66499321392421, −10.92645156824531, −10.37956717828454, −9.870745834661597, −9.329143326932995, −8.877158470576194, −8.348760504019450, −7.573895149574511, −7.382083370923344, −6.514353477055941, −6.130070186599332, −5.377435857775248, −4.666505356237472, −4.153016292031199, −3.503767228374011, −2.722086642168722, −2.355937227138102, −1.451068315565924, −0.5122983020161857, 0.5122983020161857, 1.451068315565924, 2.355937227138102, 2.722086642168722, 3.503767228374011, 4.153016292031199, 4.666505356237472, 5.377435857775248, 6.130070186599332, 6.514353477055941, 7.382083370923344, 7.573895149574511, 8.348760504019450, 8.877158470576194, 9.329143326932995, 9.870745834661597, 10.37956717828454, 10.92645156824531, 11.66499321392421, 11.94810874435642, 12.64237604059996, 13.24830747817822, 13.63016165380054, 13.94118175909205, 14.57298686440132

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