Properties

Label 2-51600-1.1-c1-0-12
Degree $2$
Conductor $51600$
Sign $1$
Analytic cond. $412.028$
Root an. cond. $20.2984$
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 + 9-s − 4·11-s − 6·17-s + 2·19-s + 6·23-s − 27-s + 4·29-s + 4·31-s + 4·33-s − 6·37-s + 10·41-s − 43-s + 6·47-s − 7·49-s + 6·51-s + 6·53-s − 2·57-s + 4·59-s + 10·61-s − 8·67-s − 6·69-s − 12·71-s − 2·73-s − 4·79-s + 81-s − 2·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.45·17-s + 0.458·19-s + 1.25·23-s − 0.192·27-s + 0.742·29-s + 0.718·31-s + 0.696·33-s − 0.986·37-s + 1.56·41-s − 0.152·43-s + 0.875·47-s − 49-s + 0.840·51-s + 0.824·53-s − 0.264·57-s + 0.520·59-s + 1.28·61-s − 0.977·67-s − 0.722·69-s − 1.42·71-s − 0.234·73-s − 0.450·79-s + 1/9·81-s − 0.219·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(412.028\)
Root analytic conductor: \(20.2984\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{51600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 51600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.299660260\)
\(L(\frac12)\) \(\approx\) \(1.299660260\)
\(L(\frac{3}{2})\) 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 \( 1 \)
43 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−14.52737449879728, −13.85599696070707, −13.30827550057160, −13.03058914364632, −12.51875462683073, −11.81939292872683, −11.41708630196679, −10.85253056288458, −10.43182065507169, −10.02386325686156, −9.208666548816613, −8.798669017850171, −8.212127486258763, −7.539366132192867, −7.011762803042543, −6.603061500681006, −5.797530713649422, −5.409974152229180, −4.637329805031433, −4.446893324969815, −3.427412186104539, −2.720205884224868, −2.241069673637545, −1.214945010375552, −0.4441442126352528, 0.4441442126352528, 1.214945010375552, 2.241069673637545, 2.720205884224868, 3.427412186104539, 4.446893324969815, 4.637329805031433, 5.409974152229180, 5.797530713649422, 6.603061500681006, 7.011762803042543, 7.539366132192867, 8.212127486258763, 8.798669017850171, 9.208666548816613, 10.02386325686156, 10.43182065507169, 10.85253056288458, 11.41708630196679, 11.81939292872683, 12.51875462683073, 13.03058914364632, 13.30827550057160, 13.85599696070707, 14.52737449879728

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