Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 4.11·5-s + 9·9-s − 17.9·11-s + 23.4·13-s − 12.3·15-s + 76.2·17-s − 35.5·19-s + 40.7·23-s − 108.·25-s + 27·27-s − 178.·29-s − 31.6·31-s − 53.8·33-s − 54.8·37-s + 70.2·39-s − 190.·41-s + 131.·43-s − 37.0·45-s − 199.·47-s + 228.·51-s + 321.·53-s + 73.8·55-s − 106.·57-s − 163.·59-s + 265.·61-s − 96.4·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.368·5-s + 0.333·9-s − 0.491·11-s + 0.499·13-s − 0.212·15-s + 1.08·17-s − 0.429·19-s + 0.369·23-s − 0.864·25-s + 0.192·27-s − 1.14·29-s − 0.183·31-s − 0.283·33-s − 0.243·37-s + 0.288·39-s − 0.725·41-s + 0.468·43-s − 0.122·45-s − 0.619·47-s + 0.627·51-s + 0.834·53-s + 0.181·55-s − 0.248·57-s − 0.360·59-s + 0.556·61-s − 0.184·65-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: \(1\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.11T + 125T^{2} \)
11 \( 1 + 17.9T + 1.33e3T^{2} \)
13 \( 1 - 23.4T + 2.19e3T^{2} \)
17 \( 1 - 76.2T + 4.91e3T^{2} \)
19 \( 1 + 35.5T + 6.85e3T^{2} \)
23 \( 1 - 40.7T + 1.21e4T^{2} \)
29 \( 1 + 178.T + 2.43e4T^{2} \)
31 \( 1 + 31.6T + 2.97e4T^{2} \)
37 \( 1 + 54.8T + 5.06e4T^{2} \)
41 \( 1 + 190.T + 6.89e4T^{2} \)
43 \( 1 - 131.T + 7.95e4T^{2} \)
47 \( 1 + 199.T + 1.03e5T^{2} \)
53 \( 1 - 321.T + 1.48e5T^{2} \)
59 \( 1 + 163.T + 2.05e5T^{2} \)
61 \( 1 - 265.T + 2.26e5T^{2} \)
67 \( 1 + 278.T + 3.00e5T^{2} \)
71 \( 1 - 10.5T + 3.57e5T^{2} \)
73 \( 1 - 584.T + 3.89e5T^{2} \)
79 \( 1 - 183.T + 4.93e5T^{2} \)
83 \( 1 - 175.T + 5.71e5T^{2} \)
89 \( 1 - 47.1T + 7.04e5T^{2} \)
97 \( 1 + 556.T + 9.12e5T^{2} \)
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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

−8.140402375839293786415808568597, −7.67631964763714001449649273675, −6.85381900336772414548878657839, −5.84231163429599317515092400655, −5.09514815667377460172346453153, −3.96684409416604842116762658666, −3.41039046916349511291395381925, −2.35789718480854725632291779520, −1.31498927224312596647926966662, 0, 1.31498927224312596647926966662, 2.35789718480854725632291779520, 3.41039046916349511291395381925, 3.96684409416604842116762658666, 5.09514815667377460172346453153, 5.84231163429599317515092400655, 6.85381900336772414548878657839, 7.67631964763714001449649273675, 8.140402375839293786415808568597

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