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 + 19.8·5-s + 9·9-s − 23.9·11-s + 87.3·13-s + 59.6·15-s − 5.63·17-s − 64.8·19-s + 25.5·23-s + 270.·25-s + 27·27-s + 60.3·29-s + 122.·31-s − 71.8·33-s − 56.1·37-s + 262.·39-s − 299.·41-s + 501.·43-s + 179.·45-s − 305.·47-s − 16.9·51-s − 375.·53-s − 476.·55-s − 194.·57-s + 627.·59-s + 3.75·61-s + 1.73e3·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.77·5-s + 0.333·9-s − 0.656·11-s + 1.86·13-s + 1.02·15-s − 0.0804·17-s − 0.783·19-s + 0.232·23-s + 2.16·25-s + 0.192·27-s + 0.386·29-s + 0.710·31-s − 0.378·33-s − 0.249·37-s + 1.07·39-s − 1.14·41-s + 1.77·43-s + 0.593·45-s − 0.948·47-s − 0.0464·51-s − 0.972·53-s − 1.16·55-s − 0.452·57-s + 1.38·59-s + 0.00788·61-s + 3.31·65-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

\(L(2)\) \(\approx\) \(4.735833730\)
\(L(\frac12)\) \(\approx\) \(4.735833730\)
\(L(\frac{5}{2})\) 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 \( 1 - 19.8T + 125T^{2} \)
11 \( 1 + 23.9T + 1.33e3T^{2} \)
13 \( 1 - 87.3T + 2.19e3T^{2} \)
17 \( 1 + 5.63T + 4.91e3T^{2} \)
19 \( 1 + 64.8T + 6.85e3T^{2} \)
23 \( 1 - 25.5T + 1.21e4T^{2} \)
29 \( 1 - 60.3T + 2.43e4T^{2} \)
31 \( 1 - 122.T + 2.97e4T^{2} \)
37 \( 1 + 56.1T + 5.06e4T^{2} \)
41 \( 1 + 299.T + 6.89e4T^{2} \)
43 \( 1 - 501.T + 7.95e4T^{2} \)
47 \( 1 + 305.T + 1.03e5T^{2} \)
53 \( 1 + 375.T + 1.48e5T^{2} \)
59 \( 1 - 627.T + 2.05e5T^{2} \)
61 \( 1 - 3.75T + 2.26e5T^{2} \)
67 \( 1 - 813.T + 3.00e5T^{2} \)
71 \( 1 + 165.T + 3.57e5T^{2} \)
73 \( 1 - 619.T + 3.89e5T^{2} \)
79 \( 1 - 138.T + 4.93e5T^{2} \)
83 \( 1 + 621.T + 5.71e5T^{2} \)
89 \( 1 - 285.T + 7.04e5T^{2} \)
97 \( 1 + 603.T + 9.12e5T^{2} \)
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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

−8.600765404686976113795867009581, −8.205983820919049588746541901251, −6.87104891129427542044626215979, −6.28097400128542236307579102505, −5.63040398975028906856935926153, −4.74146556658921449018453982204, −3.60405455283160713942887329704, −2.64912745837785602672346448803, −1.88005445508107322124373851079, −0.999518116360043025161050530163, 0.999518116360043025161050530163, 1.88005445508107322124373851079, 2.64912745837785602672346448803, 3.60405455283160713942887329704, 4.74146556658921449018453982204, 5.63040398975028906856935926153, 6.28097400128542236307579102505, 6.87104891129427542044626215979, 8.205983820919049588746541901251, 8.600765404686976113795867009581

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