Properties

Label 2-31200-1.1-c1-0-34
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 − 4·7-s + 9-s − 2·11-s + 13-s + 2·19-s − 4·21-s + 2·23-s + 27-s + 10·29-s + 4·31-s − 2·33-s − 6·37-s + 39-s − 6·41-s − 8·43-s − 12·47-s + 9·49-s − 14·53-s + 2·57-s + 6·59-s + 2·61-s − 4·63-s + 4·67-s + 2·69-s + 14·73-s + 8·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.458·19-s − 0.872·21-s + 0.417·23-s + 0.192·27-s + 1.85·29-s + 0.718·31-s − 0.348·33-s − 0.986·37-s + 0.160·39-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 9/7·49-s − 1.92·53-s + 0.264·57-s + 0.781·59-s + 0.256·61-s − 0.503·63-s + 0.488·67-s + 0.240·69-s + 1.63·73-s + 0.911·77-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: \(1\)
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.52849752267592, −14.86638446699549, −14.18718119246940, −13.70262236201878, −13.24640251508412, −12.87306314528372, −12.23531615647794, −11.80037872757987, −10.99505719604315, −10.29644760402280, −9.982321579961734, −9.517200007865586, −8.871296771718725, −8.228041133313945, −7.918133781288574, −6.863081299487959, −6.665127837414482, −6.143498435847132, −5.066500217610349, −4.844528339122449, −3.691231246138084, −3.267722174632728, −2.850040606565135, −1.982304644879773, −0.9812496177003697, 0, 0.9812496177003697, 1.982304644879773, 2.850040606565135, 3.267722174632728, 3.691231246138084, 4.844528339122449, 5.066500217610349, 6.143498435847132, 6.665127837414482, 6.863081299487959, 7.918133781288574, 8.228041133313945, 8.871296771718725, 9.517200007865586, 9.982321579961734, 10.29644760402280, 10.99505719604315, 11.80037872757987, 12.23531615647794, 12.87306314528372, 13.24640251508412, 13.70262236201878, 14.18718119246940, 14.86638446699549, 15.52849752267592

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