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 − 13.3·5-s + 9·9-s + 1.42·11-s + 38.7·13-s + 40.0·15-s + 27.3·17-s + 65.4·19-s − 2.54·23-s + 52.9·25-s − 27·27-s − 63.7·29-s − 51.9·31-s − 4.27·33-s − 335.·37-s − 116.·39-s + 447.·41-s + 170.·43-s − 120.·45-s + 116.·47-s − 82.0·51-s + 86.3·53-s − 19.0·55-s − 196.·57-s − 380.·59-s − 199.·61-s − 517.·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.19·5-s + 0.333·9-s + 0.0390·11-s + 0.827·13-s + 0.688·15-s + 0.390·17-s + 0.790·19-s − 0.0230·23-s + 0.423·25-s − 0.192·27-s − 0.408·29-s − 0.301·31-s − 0.0225·33-s − 1.49·37-s − 0.477·39-s + 1.70·41-s + 0.604·43-s − 0.397·45-s + 0.362·47-s − 0.225·51-s + 0.223·53-s − 0.0466·55-s − 0.456·57-s − 0.840·59-s − 0.419·61-s − 0.986·65-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

\(L(2)\) \(\approx\) \(1.129346184\)
\(L(\frac12)\) \(\approx\) \(1.129346184\)
\(L(\frac{5}{2})\) 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 \( 1 + 13.3T + 125T^{2} \)
11 \( 1 - 1.42T + 1.33e3T^{2} \)
13 \( 1 - 38.7T + 2.19e3T^{2} \)
17 \( 1 - 27.3T + 4.91e3T^{2} \)
19 \( 1 - 65.4T + 6.85e3T^{2} \)
23 \( 1 + 2.54T + 1.21e4T^{2} \)
29 \( 1 + 63.7T + 2.43e4T^{2} \)
31 \( 1 + 51.9T + 2.97e4T^{2} \)
37 \( 1 + 335.T + 5.06e4T^{2} \)
41 \( 1 - 447.T + 6.89e4T^{2} \)
43 \( 1 - 170.T + 7.95e4T^{2} \)
47 \( 1 - 116.T + 1.03e5T^{2} \)
53 \( 1 - 86.3T + 1.48e5T^{2} \)
59 \( 1 + 380.T + 2.05e5T^{2} \)
61 \( 1 + 199.T + 2.26e5T^{2} \)
67 \( 1 + 951.T + 3.00e5T^{2} \)
71 \( 1 + 830.T + 3.57e5T^{2} \)
73 \( 1 + 332.T + 3.89e5T^{2} \)
79 \( 1 + 755.T + 4.93e5T^{2} \)
83 \( 1 - 15.2T + 5.71e5T^{2} \)
89 \( 1 - 1.55e3T + 7.04e5T^{2} \)
97 \( 1 - 101.T + 9.12e5T^{2} \)
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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

−8.641221665607968210887838371071, −7.53465734430278184551646489545, −7.42626952475226165234369925209, −6.20470425151722741982677864992, −5.56740401076203401322752706376, −4.54967582228770980878466344380, −3.83563345623149812238579616321, −3.05898467975307927227807335375, −1.52816769061046797491386467484, −0.51319214816375689524401751450, 0.51319214816375689524401751450, 1.52816769061046797491386467484, 3.05898467975307927227807335375, 3.83563345623149812238579616321, 4.54967582228770980878466344380, 5.56740401076203401322752706376, 6.20470425151722741982677864992, 7.42626952475226165234369925209, 7.53465734430278184551646489545, 8.641221665607968210887838371071

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