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Properties

Label 2-46200-1.1-c1-0-53
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s − 11-s − 3·13-s + 7·19-s − 21-s − 6·23-s − 27-s − 9·29-s + 33-s + 3·37-s + 3·39-s + 8·41-s − 10·43-s − 3·47-s + 49-s − 6·53-s − 7·57-s + 7·59-s + 10·61-s + 63-s + 3·67-s + 6·69-s − 8·71-s + 7·73-s − 77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 1.60·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s − 1.67·29-s + 0.174·33-s + 0.493·37-s + 0.480·39-s + 1.24·41-s − 1.52·43-s − 0.437·47-s + 1/7·49-s − 0.824·53-s − 0.927·57-s + 0.911·59-s + 1.28·61-s + 0.125·63-s + 0.366·67-s + 0.722·69-s − 0.949·71-s + 0.819·73-s − 0.113·77-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: \(1\)
Selberg data: \((2,\ 46200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.84347524502017, −14.36811821705427, −13.90628067991787, −13.20893577352525, −12.86037441434108, −12.15327434971160, −11.77447140205207, −11.29329394852160, −10.88405959704918, −10.01543427675222, −9.767454877948063, −9.337713190864503, −8.418365798770874, −7.908760701928218, −7.395790339153338, −7.010564762999547, −6.127612688244972, −5.659815122982005, −5.126547377886704, −4.650901797189914, −3.847949948423092, −3.293948316190579, −2.356180689166970, −1.801755871528328, −0.8788809242751035, 0, 0.8788809242751035, 1.801755871528328, 2.356180689166970, 3.293948316190579, 3.847949948423092, 4.650901797189914, 5.126547377886704, 5.659815122982005, 6.127612688244972, 7.010564762999547, 7.395790339153338, 7.908760701928218, 8.418365798770874, 9.337713190864503, 9.767454877948063, 10.01543427675222, 10.88405959704918, 11.29329394852160, 11.77447140205207, 12.15327434971160, 12.86037441434108, 13.20893577352525, 13.90628067991787, 14.36811821705427, 14.84347524502017

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