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 − 4.58·5-s + 9·9-s + 6.48·11-s − 45.2·13-s − 13.7·15-s − 81.5·17-s − 5.05·19-s − 106.·23-s − 103.·25-s + 27·27-s − 268.·29-s + 292.·31-s + 19.4·33-s + 114.·37-s − 135.·39-s + 161.·41-s + 471.·43-s − 41.2·45-s + 346.·47-s − 244.·51-s + 405.·53-s − 29.7·55-s − 15.1·57-s − 253.·59-s + 751.·61-s + 207.·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.410·5-s + 0.333·9-s + 0.177·11-s − 0.964·13-s − 0.236·15-s − 1.16·17-s − 0.0610·19-s − 0.963·23-s − 0.831·25-s + 0.192·27-s − 1.71·29-s + 1.69·31-s + 0.102·33-s + 0.509·37-s − 0.556·39-s + 0.615·41-s + 1.67·43-s − 0.136·45-s + 1.07·47-s − 0.671·51-s + 1.05·53-s − 0.0729·55-s − 0.0352·57-s − 0.559·59-s + 1.57·61-s + 0.395·65-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

\(L(2)\) \(\approx\) \(1.867755349\)
\(L(\frac12)\) \(\approx\) \(1.867755349\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.58T + 125T^{2} \)
11 \( 1 - 6.48T + 1.33e3T^{2} \)
13 \( 1 + 45.2T + 2.19e3T^{2} \)
17 \( 1 + 81.5T + 4.91e3T^{2} \)
19 \( 1 + 5.05T + 6.85e3T^{2} \)
23 \( 1 + 106.T + 1.21e4T^{2} \)
29 \( 1 + 268.T + 2.43e4T^{2} \)
31 \( 1 - 292.T + 2.97e4T^{2} \)
37 \( 1 - 114.T + 5.06e4T^{2} \)
41 \( 1 - 161.T + 6.89e4T^{2} \)
43 \( 1 - 471.T + 7.95e4T^{2} \)
47 \( 1 - 346.T + 1.03e5T^{2} \)
53 \( 1 - 405.T + 1.48e5T^{2} \)
59 \( 1 + 253.T + 2.05e5T^{2} \)
61 \( 1 - 751.T + 2.26e5T^{2} \)
67 \( 1 + 11.6T + 3.00e5T^{2} \)
71 \( 1 - 681.T + 3.57e5T^{2} \)
73 \( 1 + 685.T + 3.89e5T^{2} \)
79 \( 1 + 0.264T + 4.93e5T^{2} \)
83 \( 1 + 437.T + 5.71e5T^{2} \)
89 \( 1 - 58.5T + 7.04e5T^{2} \)
97 \( 1 - 1.28e3T + 9.12e5T^{2} \)
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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

−8.633317885479656665747045176627, −7.78359990256618150708105149294, −7.33111070285033705664592543350, −6.37655145753054189368115001657, −5.50245667642303507380810097949, −4.30305942888510030851443219707, −3.98662607110675946174460996874, −2.65217991889846832478146604881, −2.04550379913908738161928337724, −0.56710428894951888202403424681, 0.56710428894951888202403424681, 2.04550379913908738161928337724, 2.65217991889846832478146604881, 3.98662607110675946174460996874, 4.30305942888510030851443219707, 5.50245667642303507380810097949, 6.37655145753054189368115001657, 7.33111070285033705664592543350, 7.78359990256618150708105149294, 8.633317885479656665747045176627

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