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 − 14·5-s + 9·9-s − 4·11-s − 54·13-s − 42·15-s + 14·17-s + 92·19-s + 152·23-s + 71·25-s + 27·27-s − 106·29-s − 144·31-s − 12·33-s + 158·37-s − 162·39-s + 390·41-s + 508·43-s − 126·45-s − 528·47-s + 42·51-s + 606·53-s + 56·55-s + 276·57-s − 364·59-s − 678·61-s + 756·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.25·5-s + 1/3·9-s − 0.109·11-s − 1.15·13-s − 0.722·15-s + 0.199·17-s + 1.11·19-s + 1.37·23-s + 0.567·25-s + 0.192·27-s − 0.678·29-s − 0.834·31-s − 0.0633·33-s + 0.702·37-s − 0.665·39-s + 1.48·41-s + 1.80·43-s − 0.417·45-s − 1.63·47-s + 0.115·51-s + 1.57·53-s + 0.137·55-s + 0.641·57-s − 0.803·59-s − 1.42·61-s + 1.44·65-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 \( 1 - p T \)
7 \( 1 \)
good5 \( 1 + 14 T + p^{3} T^{2} \)
11 \( 1 + 4 T + p^{3} T^{2} \)
13 \( 1 + 54 T + p^{3} T^{2} \)
17 \( 1 - 14 T + p^{3} T^{2} \)
19 \( 1 - 92 T + p^{3} T^{2} \)
23 \( 1 - 152 T + p^{3} T^{2} \)
29 \( 1 + 106 T + p^{3} T^{2} \)
31 \( 1 + 144 T + p^{3} T^{2} \)
37 \( 1 - 158 T + p^{3} T^{2} \)
41 \( 1 - 390 T + p^{3} T^{2} \)
43 \( 1 - 508 T + p^{3} T^{2} \)
47 \( 1 + 528 T + p^{3} T^{2} \)
53 \( 1 - 606 T + p^{3} T^{2} \)
59 \( 1 + 364 T + p^{3} T^{2} \)
61 \( 1 + 678 T + p^{3} T^{2} \)
67 \( 1 + 844 T + p^{3} T^{2} \)
71 \( 1 - 8 T + p^{3} T^{2} \)
73 \( 1 - 422 T + p^{3} T^{2} \)
79 \( 1 + 384 T + p^{3} T^{2} \)
83 \( 1 + 548 T + p^{3} T^{2} \)
89 \( 1 + 1194 T + p^{3} T^{2} \)
97 \( 1 - 1502 T + p^{3} T^{2} \)
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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

−8.058018946435440175812882884886, −7.38994591747810355417693611661, −7.25906402230373360239863508439, −5.82877108374002325287728538338, −4.88436170583960733903220448022, −4.15382878160483946848362619754, −3.26140185137455311161483149323, −2.56032020426206513745262221318, −1.14399748812826999310452274977, 0, 1.14399748812826999310452274977, 2.56032020426206513745262221318, 3.26140185137455311161483149323, 4.15382878160483946848362619754, 4.88436170583960733903220448022, 5.82877108374002325287728538338, 7.25906402230373360239863508439, 7.38994591747810355417693611661, 8.058018946435440175812882884886

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