Properties

Label 2-31200-1.1-c1-0-38
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 + 2·7-s + 9-s + 13-s − 4·17-s − 2·19-s − 2·21-s + 6·23-s − 27-s − 4·31-s + 2·37-s − 39-s − 6·41-s + 4·43-s + 4·47-s − 3·49-s + 4·51-s + 10·53-s + 2·57-s − 8·59-s + 6·61-s + 2·63-s − 8·67-s − 6·69-s − 16·73-s + 81-s − 4·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.970·17-s − 0.458·19-s − 0.436·21-s + 1.25·23-s − 0.192·27-s − 0.718·31-s + 0.328·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.560·51-s + 1.37·53-s + 0.264·57-s − 1.04·59-s + 0.768·61-s + 0.251·63-s − 0.977·67-s − 0.722·69-s − 1.87·73-s + 1/9·81-s − 0.439·83-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.33344014414216, −14.86759470864374, −14.40547688972089, −13.69772092167972, −13.06099605717959, −12.91559627895309, −11.98980150583029, −11.62331292699292, −11.07761554825531, −10.64645362607930, −10.21710362995047, −9.274747229569607, −8.890397445329930, −8.375715247358984, −7.582508568945511, −7.114806192395378, −6.526501998807005, −5.870682069786274, −5.290014966304979, −4.645577533410951, −4.226867980569728, −3.383006543042102, −2.519410196178829, −1.771932039355653, −1.034290857585908, 0, 1.034290857585908, 1.771932039355653, 2.519410196178829, 3.383006543042102, 4.226867980569728, 4.645577533410951, 5.290014966304979, 5.870682069786274, 6.526501998807005, 7.114806192395378, 7.582508568945511, 8.375715247358984, 8.890397445329930, 9.274747229569607, 10.21710362995047, 10.64645362607930, 11.07761554825531, 11.62331292699292, 11.98980150583029, 12.91559627895309, 13.06099605717959, 13.69772092167972, 14.40547688972089, 14.86759470864374, 15.33344014414216

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