Properties

Degree $2$
Conductor $2352$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 7.41·5-s + 9·9-s − 10.4·11-s − 2.78·13-s − 22.2·15-s − 50.4·17-s + 125.·19-s + 182.·23-s − 70.0·25-s + 27·27-s + 156.·29-s + 139.·31-s − 31.4·33-s − 394.·37-s − 8.36·39-s − 197.·41-s − 343.·43-s − 66.7·45-s − 610.·47-s − 151.·51-s − 137.·53-s + 77.7·55-s + 375.·57-s + 589.·59-s − 247.·61-s + 20.6·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.663·5-s + 0.333·9-s − 0.287·11-s − 0.0594·13-s − 0.382·15-s − 0.719·17-s + 1.50·19-s + 1.65·23-s − 0.560·25-s + 0.192·27-s + 0.999·29-s + 0.808·31-s − 0.165·33-s − 1.75·37-s − 0.0343·39-s − 0.752·41-s − 1.21·43-s − 0.221·45-s − 1.89·47-s − 0.415·51-s − 0.356·53-s + 0.190·55-s + 0.871·57-s + 1.30·59-s − 0.518·61-s + 0.0394·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{2352} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.252967427\)
\(L(\frac12)\) \(\approx\) \(2.252967427\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
7 \( 1 \)
good5 \( 1 + 7.41T + 125T^{2} \)
11 \( 1 + 10.4T + 1.33e3T^{2} \)
13 \( 1 + 2.78T + 2.19e3T^{2} \)
17 \( 1 + 50.4T + 4.91e3T^{2} \)
19 \( 1 - 125.T + 6.85e3T^{2} \)
23 \( 1 - 182.T + 1.21e4T^{2} \)
29 \( 1 - 156.T + 2.43e4T^{2} \)
31 \( 1 - 139.T + 2.97e4T^{2} \)
37 \( 1 + 394.T + 5.06e4T^{2} \)
41 \( 1 + 197.T + 6.89e4T^{2} \)
43 \( 1 + 343.T + 7.95e4T^{2} \)
47 \( 1 + 610.T + 1.03e5T^{2} \)
53 \( 1 + 137.T + 1.48e5T^{2} \)
59 \( 1 - 589.T + 2.05e5T^{2} \)
61 \( 1 + 247.T + 2.26e5T^{2} \)
67 \( 1 - 395.T + 3.00e5T^{2} \)
71 \( 1 + 285.T + 3.57e5T^{2} \)
73 \( 1 - 997.T + 3.89e5T^{2} \)
79 \( 1 - 848.T + 4.93e5T^{2} \)
83 \( 1 + 210.T + 5.71e5T^{2} \)
89 \( 1 - 553.T + 7.04e5T^{2} \)
97 \( 1 - 903.T + 9.12e5T^{2} \)
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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

−8.464310852592366618091940798794, −8.034754157404935404986058702397, −7.07781567827630552808467759151, −6.62481349207779177984000378437, −5.20290814148075528968259055727, −4.73823510772649178941135367092, −3.50505978795658088630707467199, −3.05398772055304615381578360733, −1.82449412407694517841270931693, −0.65023772706079081117068414162, 0.65023772706079081117068414162, 1.82449412407694517841270931693, 3.05398772055304615381578360733, 3.50505978795658088630707467199, 4.73823510772649178941135367092, 5.20290814148075528968259055727, 6.62481349207779177984000378437, 7.07781567827630552808467759151, 8.034754157404935404986058702397, 8.464310852592366618091940798794

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