Properties

Degree $2$
Conductor $2352$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3.41·5-s + 9-s + 2·11-s − 2.58·13-s − 3.41·15-s + 2.24·17-s − 2.82·19-s + 7.65·23-s + 6.65·25-s + 27-s − 6.82·29-s − 1.17·31-s + 2·33-s − 4·37-s − 2.58·39-s − 6.24·41-s − 5.65·43-s − 3.41·45-s − 2.82·47-s + 2.24·51-s − 2·53-s − 6.82·55-s − 2.82·57-s − 1.17·59-s − 12.2·61-s + 8.82·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.52·5-s + 0.333·9-s + 0.603·11-s − 0.717·13-s − 0.881·15-s + 0.543·17-s − 0.648·19-s + 1.59·23-s + 1.33·25-s + 0.192·27-s − 1.26·29-s − 0.210·31-s + 0.348·33-s − 0.657·37-s − 0.414·39-s − 0.974·41-s − 0.862·43-s − 0.508·45-s − 0.412·47-s + 0.314·51-s − 0.274·53-s − 0.920·55-s − 0.374·57-s − 0.152·59-s − 1.56·61-s + 1.09·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{2352} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + 3.41T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 2.58T + 13T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 7.65T + 23T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 + 1.17T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 1.17T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 + 9.31T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 - 7.31T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + 2.58T + 97T^{2} \)
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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

−8.647400233703546582468294714320, −7.72549381241748490104788887242, −7.30565209318904281267044274378, −6.54638674299259428098893164572, −5.20620313258172193691678927143, −4.43543810155034773105262990942, −3.60934246360442805807245406123, −2.98712161165684488099306228590, −1.55507447212516279169853758229, 0, 1.55507447212516279169853758229, 2.98712161165684488099306228590, 3.60934246360442805807245406123, 4.43543810155034773105262990942, 5.20620313258172193691678927143, 6.54638674299259428098893164572, 7.30565209318904281267044274378, 7.72549381241748490104788887242, 8.647400233703546582468294714320

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