Properties

Label 2-31200-1.1-c1-0-49
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 + 7-s + 9-s + 3·11-s + 13-s − 3·17-s − 6·19-s + 21-s + 23-s + 27-s + 4·29-s − 8·31-s + 3·33-s + 3·37-s + 39-s − 3·41-s + 6·43-s − 12·47-s − 6·49-s − 3·51-s + 53-s − 6·57-s − 4·59-s + 5·61-s + 63-s + 69-s − 71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.727·17-s − 1.37·19-s + 0.218·21-s + 0.208·23-s + 0.192·27-s + 0.742·29-s − 1.43·31-s + 0.522·33-s + 0.493·37-s + 0.160·39-s − 0.468·41-s + 0.914·43-s − 1.75·47-s − 6/7·49-s − 0.420·51-s + 0.137·53-s − 0.794·57-s − 0.520·59-s + 0.640·61-s + 0.125·63-s + 0.120·69-s − 0.118·71-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 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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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

−15.19187384331036, −14.75525651658919, −14.41130390125379, −13.85357176663821, −13.20767089407032, −12.79818415693748, −12.28886765212281, −11.53439784433988, −11.03779907104714, −10.69878333236683, −9.822527998581619, −9.399401905337692, −8.749104782374403, −8.416203280901998, −7.837482157112899, −7.047002079021655, −6.565433241749976, −6.100190084407505, −5.164246158656255, −4.546345675520857, −3.988735032098337, −3.422951050976370, −2.528006466269493, −1.888582188401295, −1.223893495026446, 0, 1.223893495026446, 1.888582188401295, 2.528006466269493, 3.422951050976370, 3.988735032098337, 4.546345675520857, 5.164246158656255, 6.100190084407505, 6.565433241749976, 7.047002079021655, 7.837482157112899, 8.416203280901998, 8.749104782374403, 9.399401905337692, 9.822527998581619, 10.69878333236683, 11.03779907104714, 11.53439784433988, 12.28886765212281, 12.79818415693748, 13.20767089407032, 13.85357176663821, 14.41130390125379, 14.75525651658919, 15.19187384331036

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