Properties

Label 2-224400-1.1-c1-0-228
Degree $2$
Conductor $224400$
Sign $1$
Analytic cond. $1791.84$
Root an. cond. $42.3301$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·7-s + 9-s − 11-s + 17-s − 2·21-s − 6·23-s + 27-s − 2·29-s − 4·31-s − 33-s − 2·37-s − 6·41-s − 4·43-s − 6·47-s − 3·49-s + 51-s − 8·53-s − 8·59-s − 8·61-s − 2·63-s − 4·67-s − 6·69-s + 6·71-s − 10·73-s + 2·77-s + 6·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.242·17-s − 0.436·21-s − 1.25·23-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.174·33-s − 0.328·37-s − 0.937·41-s − 0.609·43-s − 0.875·47-s − 3/7·49-s + 0.140·51-s − 1.09·53-s − 1.04·59-s − 1.02·61-s − 0.251·63-s − 0.488·67-s − 0.722·69-s + 0.712·71-s − 1.17·73-s + 0.227·77-s + 0.675·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(1791.84\)
Root analytic conductor: \(42.3301\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{224400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 224400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.52803162121415, −12.85621487308451, −12.68002974680199, −12.00366595608408, −11.74663018918791, −10.89305538047858, −10.69578380243369, −9.923031040585778, −9.751532480538657, −9.300120609578891, −8.696864063812975, −8.203804815948151, −7.820312242586546, −7.316617041912035, −6.720464089378028, −6.320080390462867, −5.773911071733933, −5.209607583522924, −4.631467084567034, −4.035288986966837, −3.456193469019584, −3.117224703470918, −2.514414679022484, −1.731110376850962, −1.411590947804810, 0, 0, 1.411590947804810, 1.731110376850962, 2.514414679022484, 3.117224703470918, 3.456193469019584, 4.035288986966837, 4.631467084567034, 5.209607583522924, 5.773911071733933, 6.320080390462867, 6.720464089378028, 7.316617041912035, 7.820312242586546, 8.203804815948151, 8.696864063812975, 9.300120609578891, 9.751532480538657, 9.923031040585778, 10.69578380243369, 10.89305538047858, 11.74663018918791, 12.00366595608408, 12.68002974680199, 12.85621487308451, 13.52803162121415

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