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Properties

Label 2-46200-1.1-c1-0-25
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 + 3·19-s − 21-s + 23-s + 27-s − 3·29-s − 2·31-s − 33-s − 8·37-s + 4·39-s − 2·41-s + 43-s − 2·47-s + 49-s − 51-s − 11·53-s + 3·57-s + 13·59-s + 5·61-s − 63-s + 14·67-s + 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 + 0.688·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s − 0.557·29-s − 0.359·31-s − 0.174·33-s − 1.31·37-s + 0.640·39-s − 0.312·41-s + 0.152·43-s − 0.291·47-s + 1/7·49-s − 0.140·51-s − 1.51·53-s + 0.397·57-s + 1.69·59-s + 0.640·61-s − 0.125·63-s + 1.71·67-s + 0.120·69-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

\(L(1)\) \(\approx\) \(2.714038349\)
\(L(\frac12)\) \(\approx\) \(2.714038349\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−14.54416493241753, −14.09465783726999, −13.54792926336077, −13.13360196703847, −12.73603629087160, −12.09004042486562, −11.43360750296347, −10.99374292349917, −10.44672978851802, −9.810569180964599, −9.387520283066276, −8.802856304778614, −8.292279116507352, −7.868077806315958, −6.972155906770649, −6.830592061403963, −5.948381119842790, −5.419272516969953, −4.816230786936248, −3.920576160060342, −3.533451852736279, −2.968596160721522, −2.133737416932758, −1.485609840677014, −0.5705485430008289, 0.5705485430008289, 1.485609840677014, 2.133737416932758, 2.968596160721522, 3.533451852736279, 3.920576160060342, 4.816230786936248, 5.419272516969953, 5.948381119842790, 6.830592061403963, 6.972155906770649, 7.868077806315958, 8.292279116507352, 8.802856304778614, 9.387520283066276, 9.810569180964599, 10.44672978851802, 10.99374292349917, 11.43360750296347, 12.09004042486562, 12.73603629087160, 13.13360196703847, 13.54792926336077, 14.09465783726999, 14.54416493241753

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