Properties

Label 2-31200-1.1-c1-0-22
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 − 3·7-s + 9-s + 3·11-s + 13-s + 7·17-s + 8·19-s − 3·21-s − 4·23-s + 27-s − 3·29-s + 11·31-s + 3·33-s + 39-s − 2·41-s + 8·43-s + 9·47-s + 2·49-s + 7·51-s + 9·53-s + 8·57-s + 9·59-s + 61-s − 3·63-s − 5·67-s − 4·69-s − 12·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 1.69·17-s + 1.83·19-s − 0.654·21-s − 0.834·23-s + 0.192·27-s − 0.557·29-s + 1.97·31-s + 0.522·33-s + 0.160·39-s − 0.312·41-s + 1.21·43-s + 1.31·47-s + 2/7·49-s + 0.980·51-s + 1.23·53-s + 1.05·57-s + 1.17·59-s + 0.128·61-s − 0.377·63-s − 0.610·67-s − 0.481·69-s − 1.40·73-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

\(L(1)\) \(\approx\) \(3.606907016\)
\(L(\frac12)\) \(\approx\) \(3.606907016\)
\(L(\frac{3}{2})\) 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 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.04624301628963, −14.46262052956719, −14.00434336760336, −13.58596327998439, −13.14669633138554, −12.22160074241520, −12.04148857952965, −11.65093985932066, −10.58893946137829, −10.09321216199351, −9.662131509087371, −9.290605819576386, −8.650061887817196, −7.911759432257784, −7.471749786926114, −6.902438938014281, −6.151361155052708, −5.760060109128253, −5.032683085916360, −3.987806672343283, −3.685167196453286, −3.037494068310573, −2.457991712523340, −1.268712457382853, −0.8079693696848023, 0.8079693696848023, 1.268712457382853, 2.457991712523340, 3.037494068310573, 3.685167196453286, 3.987806672343283, 5.032683085916360, 5.760060109128253, 6.151361155052708, 6.902438938014281, 7.471749786926114, 7.911759432257784, 8.650061887817196, 9.290605819576386, 9.662131509087371, 10.09321216199351, 10.58893946137829, 11.65093985932066, 12.04148857952965, 12.22160074241520, 13.14669633138554, 13.58596327998439, 14.00434336760336, 14.46262052956719, 15.04624301628963

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