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 + 16.6·5-s + 9·9-s + 49.2·11-s − 64.4·13-s + 49.8·15-s − 132.·17-s − 82.0·19-s − 82.0·23-s + 150.·25-s + 27·27-s + 157.·29-s − 185.·31-s + 147.·33-s − 51.9·37-s − 193.·39-s + 49.4·41-s − 313.·43-s + 149.·45-s − 553.·47-s − 397.·51-s − 619.·53-s + 817.·55-s − 246.·57-s − 712.·59-s + 287.·61-s − 1.07e3·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.48·5-s + 0.333·9-s + 1.34·11-s − 1.37·13-s + 0.857·15-s − 1.88·17-s − 0.990·19-s − 0.743·23-s + 1.20·25-s + 0.192·27-s + 1.00·29-s − 1.07·31-s + 0.779·33-s − 0.230·37-s − 0.794·39-s + 0.188·41-s − 1.11·43-s + 0.494·45-s − 1.71·47-s − 1.09·51-s − 1.60·53-s + 2.00·55-s − 0.572·57-s − 1.57·59-s + 0.603·61-s − 2.04·65-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 \( 1 - 16.6T + 125T^{2} \)
11 \( 1 - 49.2T + 1.33e3T^{2} \)
13 \( 1 + 64.4T + 2.19e3T^{2} \)
17 \( 1 + 132.T + 4.91e3T^{2} \)
19 \( 1 + 82.0T + 6.85e3T^{2} \)
23 \( 1 + 82.0T + 1.21e4T^{2} \)
29 \( 1 - 157.T + 2.43e4T^{2} \)
31 \( 1 + 185.T + 2.97e4T^{2} \)
37 \( 1 + 51.9T + 5.06e4T^{2} \)
41 \( 1 - 49.4T + 6.89e4T^{2} \)
43 \( 1 + 313.T + 7.95e4T^{2} \)
47 \( 1 + 553.T + 1.03e5T^{2} \)
53 \( 1 + 619.T + 1.48e5T^{2} \)
59 \( 1 + 712.T + 2.05e5T^{2} \)
61 \( 1 - 287.T + 2.26e5T^{2} \)
67 \( 1 + 226.T + 3.00e5T^{2} \)
71 \( 1 + 55.3T + 3.57e5T^{2} \)
73 \( 1 + 799.T + 3.89e5T^{2} \)
79 \( 1 - 120.T + 4.93e5T^{2} \)
83 \( 1 + 857.T + 5.71e5T^{2} \)
89 \( 1 - 377.T + 7.04e5T^{2} \)
97 \( 1 - 1.26e3T + 9.12e5T^{2} \)
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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

−8.565810470551502548594312328368, −7.37459206022966011910522618997, −6.46827721600026290806898644002, −6.26288260955320306579980909000, −4.89513119663858676631966708263, −4.37102320149255210143048192190, −3.11419673252730239734338411325, −2.03445090727117358991757432236, −1.75800553960802472349861135790, 0, 1.75800553960802472349861135790, 2.03445090727117358991757432236, 3.11419673252730239734338411325, 4.37102320149255210143048192190, 4.89513119663858676631966708263, 6.26288260955320306579980909000, 6.46827721600026290806898644002, 7.37459206022966011910522618997, 8.565810470551502548594312328368

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