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Properties

Label 2-46200-1.1-c1-0-40
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 + 5·13-s + 3·17-s + 8·19-s + 21-s − 7·23-s + 27-s − 5·29-s + 7·31-s + 33-s − 4·37-s + 5·39-s + 3·41-s − 9·43-s − 2·47-s + 49-s + 3·51-s − 3·53-s + 8·57-s + 5·59-s − 9·61-s + 63-s − 12·67-s − 7·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.38·13-s + 0.727·17-s + 1.83·19-s + 0.218·21-s − 1.45·23-s + 0.192·27-s − 0.928·29-s + 1.25·31-s + 0.174·33-s − 0.657·37-s + 0.800·39-s + 0.468·41-s − 1.37·43-s − 0.291·47-s + 1/7·49-s + 0.420·51-s − 0.412·53-s + 1.05·57-s + 0.650·59-s − 1.15·61-s + 0.125·63-s − 1.46·67-s − 0.842·69-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

\(L(1)\) \(\approx\) \(4.162310740\)
\(L(\frac12)\) \(\approx\) \(4.162310740\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−14.45518239860501, −14.04205327388203, −13.68168949622601, −13.32382902365624, −12.50078658152475, −11.97751398044030, −11.57883847520697, −11.07553204992516, −10.31085505133358, −9.884124682015709, −9.422881693216223, −8.716823133968166, −8.327632165456412, −7.667892322888836, −7.445547374880921, −6.452457450562087, −6.080296069230217, −5.379979156631797, −4.798160036295116, −3.983229582273784, −3.462989325032305, −3.076548283676584, −1.996976847409217, −1.467340106096975, −0.7452298059444677, 0.7452298059444677, 1.467340106096975, 1.996976847409217, 3.076548283676584, 3.462989325032305, 3.983229582273784, 4.798160036295116, 5.379979156631797, 6.080296069230217, 6.452457450562087, 7.445547374880921, 7.667892322888836, 8.327632165456412, 8.716823133968166, 9.422881693216223, 9.884124682015709, 10.31085505133358, 11.07553204992516, 11.57883847520697, 11.97751398044030, 12.50078658152475, 13.32382902365624, 13.68168949622601, 14.04205327388203, 14.45518239860501

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