↑

Properties

Label 2-46200-1.1-c1-0-20
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s − 11-s + 2·13-s − 2·17-s + 2·19-s + 21-s − 4·23-s − 27-s − 2·29-s + 8·31-s + 33-s + 8·37-s − 2·39-s + 2·41-s + 8·43-s + 8·47-s + 49-s + 2·51-s + 6·53-s − 2·57-s + 6·59-s − 63-s − 2·67-s + 4·69-s − 12·71-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 + 0.458·19-s + 0.218·21-s − 0.834·23-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.174·33-s + 1.31·37-s − 0.320·39-s + 0.312·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.264·57-s + 0.781·59-s − 0.125·63-s − 0.244·67-s + 0.481·69-s − 1.42·71-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

\(L(1)\) \(\approx\) \(1.714020840\)
\(L(\frac12)\) \(\approx\) \(1.714020840\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.66068204935634, −13.95175206765943, −13.55879162496339, −13.06772577955278, −12.57849360981247, −11.91485178878648, −11.61636704783330, −10.96419652411557, −10.50026567917191, −9.960256342366112, −9.494897225220913, −8.806669871598206, −8.334285644716917, −7.537998670083650, −7.247248920200396, −6.407916828058843, −5.935618573915106, −5.644270526598559, −4.674611536715721, −4.270994089248076, −3.627873487007696, −2.759008445315400, −2.234696750039705, −1.185781567796621, −0.5395411681395747, 0.5395411681395747, 1.185781567796621, 2.234696750039705, 2.759008445315400, 3.627873487007696, 4.270994089248076, 4.674611536715721, 5.644270526598559, 5.935618573915106, 6.407916828058843, 7.247248920200396, 7.537998670083650, 8.334285644716917, 8.806669871598206, 9.494897225220913, 9.960256342366112, 10.50026567917191, 10.96419652411557, 11.61636704783330, 11.91485178878648, 12.57849360981247, 13.06772577955278, 13.55879162496339, 13.95175206765943, 14.66068204935634

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