Properties

Label 2-46200-1.1-c1-0-30
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 − 6·17-s − 2·19-s + 21-s + 8·23-s + 27-s − 8·29-s + 33-s − 6·37-s + 2·39-s + 8·41-s + 8·43-s + 6·47-s + 49-s − 6·51-s − 10·53-s − 2·57-s + 10·61-s + 63-s + 2·67-s + 8·69-s + 8·71-s + 6·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 1.45·17-s − 0.458·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s − 1.48·29-s + 0.174·33-s − 0.986·37-s + 0.320·39-s + 1.24·41-s + 1.21·43-s + 0.875·47-s + 1/7·49-s − 0.840·51-s − 1.37·53-s − 0.264·57-s + 1.28·61-s + 0.125·63-s + 0.244·67-s + 0.963·69-s + 0.949·71-s + 0.702·73-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: $\chi_{46200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 46200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.176723573\)
\(L(\frac12)\) \(\approx\) \(3.176723573\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.47714479437703, −14.23773417121853, −13.58184441975495, −12.98771334431846, −12.82276249478061, −12.11416711953360, −11.23493817147837, −11.00023693817101, −10.72912390896380, −9.734254779228649, −9.177603887765288, −8.946339794855879, −8.368229109642211, −7.759610481936288, −7.087996139946467, −6.757243742438973, −6.015567514792967, −5.355879725874651, −4.702217572951090, −4.067967547802444, −3.641420146238449, −2.748971695789493, −2.190904300504649, −1.493872735425377, −0.6190180101580109, 0.6190180101580109, 1.493872735425377, 2.190904300504649, 2.748971695789493, 3.641420146238449, 4.067967547802444, 4.702217572951090, 5.355879725874651, 6.015567514792967, 6.757243742438973, 7.087996139946467, 7.759610481936288, 8.368229109642211, 8.946339794855879, 9.177603887765288, 9.734254779228649, 10.72912390896380, 11.00023693817101, 11.23493817147837, 12.11416711953360, 12.82276249478061, 12.98771334431846, 13.58184441975495, 14.23773417121853, 14.47714479437703

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