Properties

Label 2-51600-1.1-c1-0-29
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s − 11-s − 5·13-s − 6·17-s + 4·21-s + 23-s − 27-s − 6·29-s + 31-s + 33-s − 8·37-s + 5·39-s + 5·41-s − 43-s − 47-s + 9·49-s + 6·51-s + 3·53-s + 59-s + 2·61-s − 4·63-s + 4·67-s − 69-s − 10·71-s + 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.38·13-s − 1.45·17-s + 0.872·21-s + 0.208·23-s − 0.192·27-s − 1.11·29-s + 0.179·31-s + 0.174·33-s − 1.31·37-s + 0.800·39-s + 0.780·41-s − 0.152·43-s − 0.145·47-s + 9/7·49-s + 0.840·51-s + 0.412·53-s + 0.130·59-s + 0.256·61-s − 0.503·63-s + 0.488·67-s − 0.120·69-s − 1.18·71-s + 0.234·73-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: \(1\)
Selberg data: \((2,\ 51600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
43 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−14.99564102543036, −14.15343503922751, −13.55530447894362, −13.09091295206134, −12.75476840121211, −12.21048421513789, −11.80147961993359, −11.04800883039349, −10.64364412228793, −10.07765436291995, −9.562941055809457, −9.203984252525366, −8.613364858136627, −7.721099170116129, −7.230904528496006, −6.702851511987075, −6.386277257945254, −5.587426285789817, −5.141313058129871, −4.460501943171615, −3.833367775732183, −3.142952915737366, −2.449556563404609, −1.903477255078759, −0.5813607124330292, 0, 0.5813607124330292, 1.903477255078759, 2.449556563404609, 3.142952915737366, 3.833367775732183, 4.460501943171615, 5.141313058129871, 5.587426285789817, 6.386277257945254, 6.702851511987075, 7.230904528496006, 7.721099170116129, 8.613364858136627, 9.203984252525366, 9.562941055809457, 10.07765436291995, 10.64364412228793, 11.04800883039349, 11.80147961993359, 12.21048421513789, 12.75476840121211, 13.09091295206134, 13.55530447894362, 14.15343503922751, 14.99564102543036

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