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Properties

Label 2-46200-1.1-c1-0-22
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 + 4·13-s + 6·17-s + 4·19-s − 21-s − 6·23-s − 27-s − 6·31-s + 33-s + 8·37-s − 4·39-s − 6·41-s − 4·43-s − 8·47-s + 49-s − 6·51-s − 6·53-s − 4·57-s − 4·59-s + 6·61-s + 63-s − 12·67-s + 6·69-s − 4·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 1.45·17-s + 0.917·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s − 1.07·31-s + 0.174·33-s + 1.31·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.768·61-s + 0.125·63-s − 1.46·67-s + 0.722·69-s − 0.468·73-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 46200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.992154149\)
\(L(\frac12)\) \(\approx\) \(1.992154149\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−14.73680112887135, −14.00051511496264, −13.64805697096541, −13.01806762881271, −12.56371539304854, −11.80981177464635, −11.65990958003935, −11.04470895009736, −10.45951399158840, −9.978151310091641, −9.526214714409981, −8.812931147265318, −8.096668844033812, −7.781383898825784, −7.253398019939771, −6.353478327487264, −6.013603302981291, −5.401987435227782, −4.951123532526119, −4.167948728861079, −3.494726594880148, −3.054685606021034, −1.876357146068595, −1.402333359047051, −0.5479117862920524, 0.5479117862920524, 1.402333359047051, 1.876357146068595, 3.054685606021034, 3.494726594880148, 4.167948728861079, 4.951123532526119, 5.401987435227782, 6.013603302981291, 6.353478327487264, 7.253398019939771, 7.781383898825784, 8.096668844033812, 8.812931147265318, 9.526214714409981, 9.978151310091641, 10.45951399158840, 11.04470895009736, 11.65990958003935, 11.80981177464635, 12.56371539304854, 13.01806762881271, 13.64805697096541, 14.00051511496264, 14.73680112887135

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