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Properties

Label 2-46200-1.1-c1-0-36
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 − 7·19-s − 21-s + 8·23-s + 27-s − 6·29-s + 33-s + 3·37-s + 5·39-s − 3·41-s + 4·43-s + 4·47-s + 49-s + 3·51-s + 7·53-s − 7·57-s + 2·59-s + 61-s − 63-s − 3·67-s + 8·69-s + 15·71-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 − 1.60·19-s − 0.218·21-s + 1.66·23-s + 0.192·27-s − 1.11·29-s + 0.174·33-s + 0.493·37-s + 0.800·39-s − 0.468·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.420·51-s + 0.961·53-s − 0.927·57-s + 0.260·59-s + 0.128·61-s − 0.125·63-s − 0.366·67-s + 0.963·69-s + 1.78·71-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

\(L(1)\) \(\approx\) \(3.438629383\)
\(L(\frac12)\) \(\approx\) \(3.438629383\)
\(L(\frac{3}{2})\) 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 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.75270639077822, −14.09356812022790, −13.45798568787942, −13.18524387906347, −12.66124901898405, −12.21250597911431, −11.32066170362232, −11.00911550666635, −10.50112136497821, −9.868900503915655, −9.248455241959847, −8.759062979961618, −8.503907579161558, −7.708225593360204, −7.195283956870208, −6.536209103439924, −6.110740844725745, −5.443458500616537, −4.724673223920004, −3.809732233559180, −3.744367020213042, −2.871137798072583, −2.204212709755762, −1.383698751716146, −0.6755101119897546, 0.6755101119897546, 1.383698751716146, 2.204212709755762, 2.871137798072583, 3.744367020213042, 3.809732233559180, 4.724673223920004, 5.443458500616537, 6.110740844725745, 6.536209103439924, 7.195283956870208, 7.708225593360204, 8.503907579161558, 8.759062979961618, 9.248455241959847, 9.868900503915655, 10.50112136497821, 11.00911550666635, 11.32066170362232, 12.21250597911431, 12.66124901898405, 13.18524387906347, 13.45798568787942, 14.09356812022790, 14.75270639077822

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