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 − 15.8·5-s + 9·9-s − 57.3·11-s + 5.69·13-s + 47.6·15-s − 51.8·17-s − 16.2·19-s + 213.·23-s + 127.·25-s − 27·27-s − 218.·29-s + 251.·31-s + 172.·33-s + 386.·37-s − 17.0·39-s − 328.·41-s + 37.5·43-s − 143.·45-s + 254.·47-s + 155.·51-s + 211.·53-s + 912.·55-s + 48.6·57-s + 412.·59-s − 836.·61-s − 90.6·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.42·5-s + 0.333·9-s − 1.57·11-s + 0.121·13-s + 0.821·15-s − 0.740·17-s − 0.195·19-s + 1.93·23-s + 1.02·25-s − 0.192·27-s − 1.39·29-s + 1.45·31-s + 0.908·33-s + 1.71·37-s − 0.0701·39-s − 1.25·41-s + 0.133·43-s − 0.474·45-s + 0.791·47-s + 0.427·51-s + 0.548·53-s + 2.23·55-s + 0.112·57-s + 0.909·59-s − 1.75·61-s − 0.172·65-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 \( 1 + 15.8T + 125T^{2} \)
11 \( 1 + 57.3T + 1.33e3T^{2} \)
13 \( 1 - 5.69T + 2.19e3T^{2} \)
17 \( 1 + 51.8T + 4.91e3T^{2} \)
19 \( 1 + 16.2T + 6.85e3T^{2} \)
23 \( 1 - 213.T + 1.21e4T^{2} \)
29 \( 1 + 218.T + 2.43e4T^{2} \)
31 \( 1 - 251.T + 2.97e4T^{2} \)
37 \( 1 - 386.T + 5.06e4T^{2} \)
41 \( 1 + 328.T + 6.89e4T^{2} \)
43 \( 1 - 37.5T + 7.95e4T^{2} \)
47 \( 1 - 254.T + 1.03e5T^{2} \)
53 \( 1 - 211.T + 1.48e5T^{2} \)
59 \( 1 - 412.T + 2.05e5T^{2} \)
61 \( 1 + 836.T + 2.26e5T^{2} \)
67 \( 1 - 165.T + 3.00e5T^{2} \)
71 \( 1 - 465.T + 3.57e5T^{2} \)
73 \( 1 - 449.T + 3.89e5T^{2} \)
79 \( 1 - 343.T + 4.93e5T^{2} \)
83 \( 1 + 1.50e3T + 5.71e5T^{2} \)
89 \( 1 + 341.T + 7.04e5T^{2} \)
97 \( 1 + 865.T + 9.12e5T^{2} \)
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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

−8.139691668642068280721660817580, −7.46731798047526092276988158347, −6.89176886707191456339329952219, −5.82953474884432099329149565301, −4.92638340205927121163499983070, −4.38825985099628720635506825634, −3.34344178470066551648020145337, −2.44926870063739286823686605119, −0.845022109851300899617793417665, 0, 0.845022109851300899617793417665, 2.44926870063739286823686605119, 3.34344178470066551648020145337, 4.38825985099628720635506825634, 4.92638340205927121163499983070, 5.82953474884432099329149565301, 6.89176886707191456339329952219, 7.46731798047526092276988158347, 8.139691668642068280721660817580

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