Properties

Label 2-31200-1.1-c1-0-2
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 − 5·11-s − 13-s + 5·17-s − 4·19-s − 3·21-s + 2·23-s − 27-s − 9·29-s − 3·31-s + 5·33-s + 10·37-s + 39-s − 12·41-s + 2·43-s + 9·47-s + 2·49-s − 5·51-s − 9·53-s + 4·57-s − 3·59-s − 7·61-s + 3·63-s + 9·67-s − 2·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 1.21·17-s − 0.917·19-s − 0.654·21-s + 0.417·23-s − 0.192·27-s − 1.67·29-s − 0.538·31-s + 0.870·33-s + 1.64·37-s + 0.160·39-s − 1.87·41-s + 0.304·43-s + 1.31·47-s + 2/7·49-s − 0.700·51-s − 1.23·53-s + 0.529·57-s − 0.390·59-s − 0.896·61-s + 0.377·63-s + 1.09·67-s − 0.240·69-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

\(L(1)\) \(\approx\) \(1.336698413\)
\(L(\frac12)\) \(\approx\) \(1.336698413\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−14.97046412778789, −14.77552410759676, −14.08545027648177, −13.32629031028548, −13.04349891660546, −12.31401499387322, −12.02405297136829, −11.12340729376069, −10.87591334329287, −10.55188145703307, −9.637855622469574, −9.366305493512541, −8.222527795030437, −8.060247101845268, −7.513670334829015, −6.912309088879161, −6.045946117149182, −5.422699778606698, −5.156893747320645, −4.503282674557091, −3.769601374562376, −2.912564883476455, −2.131843540524888, −1.495966581308414, −0.4556841741670643, 0.4556841741670643, 1.495966581308414, 2.131843540524888, 2.912564883476455, 3.769601374562376, 4.503282674557091, 5.156893747320645, 5.422699778606698, 6.045946117149182, 6.912309088879161, 7.513670334829015, 8.060247101845268, 8.222527795030437, 9.366305493512541, 9.637855622469574, 10.55188145703307, 10.87591334329287, 11.12340729376069, 12.02405297136829, 12.31401499387322, 13.04349891660546, 13.32629031028548, 14.08545027648177, 14.77552410759676, 14.97046412778789

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