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 + 18·5-s + 9·9-s + 50·11-s − 36·13-s − 54·15-s + 126·17-s + 72·19-s − 14·23-s + 199·25-s − 27·27-s + 158·29-s + 36·31-s − 150·33-s − 162·37-s + 108·39-s − 270·41-s + 324·43-s + 162·45-s + 72·47-s − 378·51-s − 22·53-s + 900·55-s − 216·57-s − 468·59-s + 792·61-s − 648·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.60·5-s + 1/3·9-s + 1.37·11-s − 0.768·13-s − 0.929·15-s + 1.79·17-s + 0.869·19-s − 0.126·23-s + 1.59·25-s − 0.192·27-s + 1.01·29-s + 0.208·31-s − 0.791·33-s − 0.719·37-s + 0.443·39-s − 1.02·41-s + 1.14·43-s + 0.536·45-s + 0.223·47-s − 1.03·51-s − 0.0570·53-s + 2.20·55-s − 0.501·57-s − 1.03·59-s + 1.66·61-s − 1.23·65-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

\(L(2)\) \(\approx\) \(3.352654229\)
\(L(\frac12)\) \(\approx\) \(3.352654229\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
7 \( 1 \)
good5 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 - 50 T + p^{3} T^{2} \)
13 \( 1 + 36 T + p^{3} T^{2} \)
17 \( 1 - 126 T + p^{3} T^{2} \)
19 \( 1 - 72 T + p^{3} T^{2} \)
23 \( 1 + 14 T + p^{3} T^{2} \)
29 \( 1 - 158 T + p^{3} T^{2} \)
31 \( 1 - 36 T + p^{3} T^{2} \)
37 \( 1 + 162 T + p^{3} T^{2} \)
41 \( 1 + 270 T + p^{3} T^{2} \)
43 \( 1 - 324 T + p^{3} T^{2} \)
47 \( 1 - 72 T + p^{3} T^{2} \)
53 \( 1 + 22 T + p^{3} T^{2} \)
59 \( 1 + 468 T + p^{3} T^{2} \)
61 \( 1 - 792 T + p^{3} T^{2} \)
67 \( 1 + 232 T + p^{3} T^{2} \)
71 \( 1 - 734 T + p^{3} T^{2} \)
73 \( 1 - 180 T + p^{3} T^{2} \)
79 \( 1 + 236 T + p^{3} T^{2} \)
83 \( 1 + 36 T + p^{3} T^{2} \)
89 \( 1 - 234 T + p^{3} T^{2} \)
97 \( 1 - 468 T + p^{3} T^{2} \)
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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

−8.873977219687825779919825915596, −7.74877816741024729473368980764, −6.85337438573413902160064928986, −6.26233257842090436315388472417, −5.48384694576017719031767630304, −5.00007029092371776677941909778, −3.75705803823355954788040534180, −2.70663974270196834687652420679, −1.56788559189220745891005266185, −0.938063913897322390962744523815, 0.938063913897322390962744523815, 1.56788559189220745891005266185, 2.70663974270196834687652420679, 3.75705803823355954788040534180, 5.00007029092371776677941909778, 5.48384694576017719031767630304, 6.26233257842090436315388472417, 6.85337438573413902160064928986, 7.74877816741024729473368980764, 8.873977219687825779919825915596

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