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Properties

Label 2-46200-1.1-c1-0-19
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 + 2·13-s + 2·17-s + 4·19-s + 21-s − 27-s + 2·29-s − 8·31-s − 33-s + 2·37-s − 2·39-s − 10·41-s + 12·43-s + 49-s − 2·51-s − 2·53-s − 4·57-s − 6·61-s − 63-s + 8·67-s − 8·71-s − 2·73-s − 77-s + 4·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.218·21-s − 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.174·33-s + 0.328·37-s − 0.320·39-s − 1.56·41-s + 1.82·43-s + 1/7·49-s − 0.280·51-s − 0.274·53-s − 0.529·57-s − 0.768·61-s − 0.125·63-s + 0.977·67-s − 0.949·71-s − 0.234·73-s − 0.113·77-s + 0.450·79-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

\(L(1)\) \(\approx\) \(1.796384519\)
\(L(\frac12)\) \(\approx\) \(1.796384519\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.58078692578284, −14.04484844365549, −13.61549348409130, −12.98822542116836, −12.51795901126702, −12.04193113987452, −11.53514290003284, −10.97047692783448, −10.57289091338702, −9.866783873630814, −9.476427107104439, −8.910460819262807, −8.285134808943007, −7.578320388874666, −7.132902674445119, −6.567928134229699, −5.830586223749887, −5.606204479190966, −4.829243714450068, −4.177698627814923, −3.491799300727041, −3.033218212336686, −2.011770781836719, −1.282677076631566, −0.5360102395858623, 0.5360102395858623, 1.282677076631566, 2.011770781836719, 3.033218212336686, 3.491799300727041, 4.177698627814923, 4.829243714450068, 5.606204479190966, 5.830586223749887, 6.567928134229699, 7.132902674445119, 7.578320388874666, 8.285134808943007, 8.910460819262807, 9.476427107104439, 9.866783873630814, 10.57289091338702, 10.97047692783448, 11.53514290003284, 12.04193113987452, 12.51795901126702, 12.98822542116836, 13.61549348409130, 14.04484844365549, 14.58078692578284

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