Properties

Label 2-31200-1.1-c1-0-32
Degree $2$
Conductor $31200$
Sign $-1$
Analytic cond. $249.133$
Root an. cond. $15.7839$
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 − 3·7-s + 9-s − 3·11-s − 13-s + 3·17-s + 2·19-s − 3·21-s + 23-s + 27-s − 8·29-s + 4·31-s − 3·33-s + 5·37-s − 39-s − 7·41-s − 2·43-s + 4·47-s + 2·49-s + 3·51-s + 11·53-s + 2·57-s − 4·59-s + 61-s − 3·63-s + 69-s + 5·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 0.727·17-s + 0.458·19-s − 0.654·21-s + 0.208·23-s + 0.192·27-s − 1.48·29-s + 0.718·31-s − 0.522·33-s + 0.821·37-s − 0.160·39-s − 1.09·41-s − 0.304·43-s + 0.583·47-s + 2/7·49-s + 0.420·51-s + 1.51·53-s + 0.264·57-s − 0.520·59-s + 0.128·61-s − 0.377·63-s + 0.120·69-s + 0.593·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31200\)    =    \(2^{5} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(249.133\)
Root analytic conductor: \(15.7839\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{31200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−15.28309930574313, −14.92698987376217, −14.32534809618907, −13.50675091922193, −13.40276341548179, −12.81974767444786, −12.26141925343099, −11.76479690093328, −11.00666878472858, −10.33780951271133, −9.970453712853755, −9.430094400645800, −9.015098219737754, −8.133365401312386, −7.821916317758713, −7.105471326622568, −6.690302865984211, −5.796514668128037, −5.419114908732734, −4.632510099305973, −3.801526692268234, −3.276742706785232, −2.725638978018236, −2.046465102524166, −0.9764412989421809, 0, 0.9764412989421809, 2.046465102524166, 2.725638978018236, 3.276742706785232, 3.801526692268234, 4.632510099305973, 5.419114908732734, 5.796514668128037, 6.690302865984211, 7.105471326622568, 7.821916317758713, 8.133365401312386, 9.015098219737754, 9.430094400645800, 9.970453712853755, 10.33780951271133, 11.00666878472858, 11.76479690093328, 12.26141925343099, 12.81974767444786, 13.40276341548179, 13.50675091922193, 14.32534809618907, 14.92698987376217, 15.28309930574313

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