Properties

Degree $2$
Conductor $133200$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·11-s + 2·13-s + 2·17-s − 4·19-s + 8·23-s + 2·29-s − 8·31-s − 37-s − 10·41-s + 12·43-s − 7·49-s + 6·53-s + 4·59-s − 10·61-s − 4·67-s + 8·71-s + 6·73-s + 8·79-s + 4·83-s − 10·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s + 0.371·29-s − 1.43·31-s − 0.164·37-s − 1.56·41-s + 1.82·43-s − 49-s + 0.824·53-s + 0.520·59-s − 1.28·61-s − 0.488·67-s + 0.949·71-s + 0.702·73-s + 0.900·79-s + 0.439·83-s − 1.05·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 37\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{133200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 133200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
37 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.52535613836821, −13.38945224654418, −12.58128873614222, −12.37549322529983, −11.82012007334898, −11.18231105039629, −10.86244148720128, −10.50039677554292, −9.715839699710807, −9.161551940200149, −9.032676736108392, −8.353719852357292, −7.899498690371934, −7.159815371033053, −6.733581507780580, −6.419219330918269, −5.651315108355289, −5.255131181077501, −4.566728387336953, −3.961480993704010, −3.560005118209722, −2.939364839303437, −2.191512464033444, −1.433530277856061, −1.020997274473000, 0, 1.020997274473000, 1.433530277856061, 2.191512464033444, 2.939364839303437, 3.560005118209722, 3.961480993704010, 4.566728387336953, 5.255131181077501, 5.651315108355289, 6.419219330918269, 6.733581507780580, 7.159815371033053, 7.899498690371934, 8.353719852357292, 9.032676736108392, 9.161551940200149, 9.715839699710807, 10.50039677554292, 10.86244148720128, 11.18231105039629, 11.82012007334898, 12.37549322529983, 12.58128873614222, 13.38945224654418, 13.52535613836821

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