Properties

Label 2-51600-1.1-c1-0-3
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 − 3·11-s + 13-s + 6·17-s − 4·19-s − 7·23-s − 27-s + 2·29-s − 5·31-s + 3·33-s − 4·37-s − 39-s + 41-s − 43-s − 9·47-s − 7·49-s − 6·51-s − 11·53-s + 4·57-s − 5·59-s + 6·61-s − 4·67-s + 7·69-s + 6·71-s − 6·73-s + 3·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 1.45·17-s − 0.917·19-s − 1.45·23-s − 0.192·27-s + 0.371·29-s − 0.898·31-s + 0.522·33-s − 0.657·37-s − 0.160·39-s + 0.156·41-s − 0.152·43-s − 1.31·47-s − 49-s − 0.840·51-s − 1.51·53-s + 0.529·57-s − 0.650·59-s + 0.768·61-s − 0.488·67-s + 0.842·69-s + 0.712·71-s − 0.702·73-s + 0.337·79-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

\(L(1)\) \(\approx\) \(0.6426178457\)
\(L(\frac12)\) \(\approx\) \(0.6426178457\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−14.34545251572309, −14.13197042197513, −13.37487993762679, −12.77269030246073, −12.54264040483611, −11.97106578405395, −11.35873584637046, −10.90082698961940, −10.35388106549604, −9.903832536063213, −9.521663377089972, −8.581403776864799, −8.114070586251727, −7.753156760773801, −7.045752596280863, −6.400174047385213, −5.895611098041743, −5.404546431879201, −4.826787000927082, −4.179497822997197, −3.471472454226024, −2.914063425887484, −1.938497775357747, −1.442203237579806, −0.2867342014930408, 0.2867342014930408, 1.442203237579806, 1.938497775357747, 2.914063425887484, 3.471472454226024, 4.179497822997197, 4.826787000927082, 5.404546431879201, 5.895611098041743, 6.400174047385213, 7.045752596280863, 7.753156760773801, 8.114070586251727, 8.581403776864799, 9.521663377089972, 9.903832536063213, 10.35388106549604, 10.90082698961940, 11.35873584637046, 11.97106578405395, 12.54264040483611, 12.77269030246073, 13.37487993762679, 14.13197042197513, 14.34545251572309

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