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 + 12.8·5-s + 9·9-s + 36.8·11-s − 87.1·13-s − 38.4·15-s − 102.·17-s + 95.8·19-s + 96·23-s + 39.4·25-s − 27·27-s − 212.·29-s − 159.·31-s − 110.·33-s + 128.·37-s + 261.·39-s + 298.·41-s + 33.3·43-s + 115.·45-s + 271.·47-s + 307.·51-s + 448.·53-s + 472.·55-s − 287.·57-s − 668.·59-s + 243.·61-s − 1.11e3·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.14·5-s + 0.333·9-s + 1.00·11-s − 1.85·13-s − 0.662·15-s − 1.46·17-s + 1.15·19-s + 0.870·23-s + 0.315·25-s − 0.192·27-s − 1.35·29-s − 0.922·31-s − 0.582·33-s + 0.571·37-s + 1.07·39-s + 1.13·41-s + 0.118·43-s + 0.382·45-s + 0.841·47-s + 0.845·51-s + 1.16·53-s + 1.15·55-s − 0.668·57-s − 1.47·59-s + 0.511·61-s − 2.13·65-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

\(L(2)\) \(\approx\) \(2.022440010\)
\(L(\frac12)\) \(\approx\) \(2.022440010\)
\(L(\frac{5}{2})\) 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 \( 1 - 12.8T + 125T^{2} \)
11 \( 1 - 36.8T + 1.33e3T^{2} \)
13 \( 1 + 87.1T + 2.19e3T^{2} \)
17 \( 1 + 102.T + 4.91e3T^{2} \)
19 \( 1 - 95.8T + 6.85e3T^{2} \)
23 \( 1 - 96T + 1.21e4T^{2} \)
29 \( 1 + 212.T + 2.43e4T^{2} \)
31 \( 1 + 159.T + 2.97e4T^{2} \)
37 \( 1 - 128.T + 5.06e4T^{2} \)
41 \( 1 - 298.T + 6.89e4T^{2} \)
43 \( 1 - 33.3T + 7.95e4T^{2} \)
47 \( 1 - 271.T + 1.03e5T^{2} \)
53 \( 1 - 448.T + 1.48e5T^{2} \)
59 \( 1 + 668.T + 2.05e5T^{2} \)
61 \( 1 - 243.T + 2.26e5T^{2} \)
67 \( 1 - 335.T + 3.00e5T^{2} \)
71 \( 1 - 339.T + 3.57e5T^{2} \)
73 \( 1 - 918.T + 3.89e5T^{2} \)
79 \( 1 - 136.T + 4.93e5T^{2} \)
83 \( 1 - 287.T + 5.71e5T^{2} \)
89 \( 1 + 161.T + 7.04e5T^{2} \)
97 \( 1 + 182.T + 9.12e5T^{2} \)
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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

−9.103464380242617553511098400996, −7.54629153979818213215710048868, −7.06807268117595668821653741597, −6.25995647612891808886014689614, −5.47227013401897378005324302583, −4.86869187155447961250320973521, −3.90041283273099290310489182724, −2.54251591136882356422614091910, −1.84786725164573759329089512512, −0.64608099154644419861153798470, 0.64608099154644419861153798470, 1.84786725164573759329089512512, 2.54251591136882356422614091910, 3.90041283273099290310489182724, 4.86869187155447961250320973521, 5.47227013401897378005324302583, 6.25995647612891808886014689614, 7.06807268117595668821653741597, 7.54629153979818213215710048868, 9.103464380242617553511098400996

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