Properties

Degree $2$
Conductor $331200$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s − 2·13-s − 6·17-s + 4·19-s + 23-s − 2·29-s − 2·37-s − 10·41-s + 4·43-s − 7·49-s − 6·53-s + 4·59-s + 10·61-s + 12·67-s − 8·71-s − 10·73-s + 8·79-s − 4·83-s − 18·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.20·11-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.208·23-s − 0.371·29-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 49-s − 0.824·53-s + 0.520·59-s + 1.28·61-s + 1.46·67-s − 0.949·71-s − 1.17·73-s + 0.900·79-s − 0.439·83-s − 1.90·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(331200\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{331200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 331200,\ (\ :1/2),\ 1)\)

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 \( 1 \)
23 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−12.95931464462819, −12.76246807050348, −12.17086500566622, −11.54324311723776, −11.29043710633501, −10.85704637547706, −10.22930360319834, −9.917958684793024, −9.516821760470729, −8.741992115499928, −8.642235342566732, −7.877501120369636, −7.601271786185495, −6.978026900875380, −6.677712405201763, −6.052643501082044, −5.371401829006638, −5.070444906934118, −4.710204413235839, −3.941892766988918, −3.506380133171778, −2.666190283811162, −2.550140393029499, −1.775623581903788, −1.143968169334409, 0, 0, 1.143968169334409, 1.775623581903788, 2.550140393029499, 2.666190283811162, 3.506380133171778, 3.941892766988918, 4.710204413235839, 5.070444906934118, 5.371401829006638, 6.052643501082044, 6.677712405201763, 6.978026900875380, 7.601271786185495, 7.877501120369636, 8.642235342566732, 8.741992115499928, 9.516821760470729, 9.917958684793024, 10.22930360319834, 10.85704637547706, 11.29043710633501, 11.54324311723776, 12.17086500566622, 12.76246807050348, 12.95931464462819

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