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Properties

Label 2-46200-1.1-c1-0-16
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 + 2·37-s + 2·39-s − 10·41-s − 4·43-s + 49-s − 2·51-s + 6·53-s − 12·59-s − 2·61-s + 63-s − 4·67-s − 4·69-s + 2·73-s − 77-s + 16·79-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 + 0.328·37-s + 0.320·39-s − 1.56·41-s − 0.609·43-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 1.56·59-s − 0.256·61-s + 0.125·63-s − 0.488·67-s − 0.481·69-s + 0.234·73-s − 0.113·77-s + 1.80·79-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

\(L(1)\) \(\approx\) \(1.794957342\)
\(L(\frac12)\) \(\approx\) \(1.794957342\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−14.74845471279849, −14.02729230414401, −13.54048026731872, −13.15555759479315, −12.27071474639304, −12.12283270534118, −11.63937447095380, −10.87083478240782, −10.55908084353504, −9.994309929641876, −9.492699519754072, −8.798012475495239, −8.214261788726615, −7.716328670135410, −7.115567316643602, −6.536773405910533, −6.050491733193915, −5.171359166469411, −4.961622649800767, −4.373062557532863, −3.462290213235253, −2.889745829417104, −2.093751456172223, −1.281398479726226, −0.5308190413782425, 0.5308190413782425, 1.281398479726226, 2.093751456172223, 2.889745829417104, 3.462290213235253, 4.373062557532863, 4.961622649800767, 5.171359166469411, 6.050491733193915, 6.536773405910533, 7.115567316643602, 7.716328670135410, 8.214261788726615, 8.798012475495239, 9.492699519754072, 9.994309929641876, 10.55908084353504, 10.87083478240782, 11.63937447095380, 12.12283270534118, 12.27071474639304, 13.15555759479315, 13.54048026731872, 14.02729230414401, 14.74845471279849

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