Properties

Label 2-31200-1.1-c1-0-19
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4·7-s + 9-s − 4·11-s + 13-s − 2·17-s + 4·19-s + 4·21-s + 27-s + 6·29-s − 4·33-s − 10·37-s + 39-s + 2·41-s − 4·43-s + 4·47-s + 9·49-s − 2·51-s + 6·53-s + 4·57-s + 4·59-s − 2·61-s + 4·63-s − 4·67-s + 8·71-s + 2·73-s − 16·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.485·17-s + 0.917·19-s + 0.872·21-s + 0.192·27-s + 1.11·29-s − 0.696·33-s − 1.64·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s + 0.520·59-s − 0.256·61-s + 0.503·63-s − 0.488·67-s + 0.949·71-s + 0.234·73-s − 1.82·77-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: \(0\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.557757255\)
\(L(\frac12)\) \(\approx\) \(3.557757255\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.25251425883895, −14.36122779141423, −14.16509318186877, −13.50625223140735, −13.24278927529726, −12.30863010027744, −12.00466852633983, −11.23574666923711, −10.87844062956161, −10.23212800673204, −9.849751965367334, −8.820382330710134, −8.597922564143234, −8.083602228981987, −7.437511457438112, −7.131361558823362, −6.179015603996052, −5.382378278352650, −4.984334190783072, −4.451544398365686, −3.627512842514614, −2.904755934774069, −2.217430175688605, −1.600887400731496, −0.7046322021598749, 0.7046322021598749, 1.600887400731496, 2.217430175688605, 2.904755934774069, 3.627512842514614, 4.451544398365686, 4.984334190783072, 5.382378278352650, 6.179015603996052, 7.131361558823362, 7.437511457438112, 8.083602228981987, 8.597922564143234, 8.820382330710134, 9.849751965367334, 10.23212800673204, 10.87844062956161, 11.23574666923711, 12.00466852633983, 12.30863010027744, 13.24278927529726, 13.50625223140735, 14.16509318186877, 14.36122779141423, 15.25251425883895

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