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Properties

Label 2-46200-1.1-c1-0-39
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 + 4·13-s − 17-s + 5·19-s + 21-s − 9·23-s + 27-s + 9·29-s + 2·31-s − 33-s + 8·37-s + 4·39-s − 2·41-s + 11·43-s − 2·47-s + 49-s − 51-s + 3·53-s + 5·57-s − 59-s + 11·61-s + 63-s − 6·67-s − 9·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.242·17-s + 1.14·19-s + 0.218·21-s − 1.87·23-s + 0.192·27-s + 1.67·29-s + 0.359·31-s − 0.174·33-s + 1.31·37-s + 0.640·39-s − 0.312·41-s + 1.67·43-s − 0.291·47-s + 1/7·49-s − 0.140·51-s + 0.412·53-s + 0.662·57-s − 0.130·59-s + 1.40·61-s + 0.125·63-s − 0.733·67-s − 1.08·69-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

\(L(1)\) \(\approx\) \(3.837158257\)
\(L(\frac12)\) \(\approx\) \(3.837158257\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.47523876374400, −14.08600980906108, −13.60145469772471, −13.33179424737237, −12.48335582064978, −12.11279863320192, −11.45226828023579, −11.08837811930855, −10.23649036253068, −10.04782861519681, −9.346934921694462, −8.729808310095162, −8.232089934573014, −7.877935674898815, −7.286645685744665, −6.542835833026173, −5.973723030361870, −5.505450899254935, −4.576352443481694, −4.203847717307245, −3.517157161054717, −2.802243255735576, −2.243877461831765, −1.370852316712585, −0.7217275580251307, 0.7217275580251307, 1.370852316712585, 2.243877461831765, 2.802243255735576, 3.517157161054717, 4.203847717307245, 4.576352443481694, 5.505450899254935, 5.973723030361870, 6.542835833026173, 7.286645685744665, 7.877935674898815, 8.232089934573014, 8.729808310095162, 9.346934921694462, 10.04782861519681, 10.23649036253068, 11.08837811930855, 11.45226828023579, 12.11279863320192, 12.48335582064978, 13.33179424737237, 13.60145469772471, 14.08600980906108, 14.47523876374400

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