Properties

Degree $2$
Conductor $3528$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.27·5-s + 4.27·11-s + 1.27·13-s − 4·17-s − 1.27·19-s − 4·23-s + 13.2·25-s + 2.27·29-s + 31-s + 5.27·37-s + 10.5·41-s − 7.27·43-s − 6·47-s − 1.72·53-s − 18.2·55-s + 6.27·59-s − 10·61-s − 5.45·65-s + 7.27·67-s − 2·71-s − 3.27·73-s − 3.54·79-s − 0.274·83-s + 17.0·85-s + 4.54·89-s + 5.45·95-s − 16.2·97-s + ⋯
L(s)  = 1  − 1.91·5-s + 1.28·11-s + 0.353·13-s − 0.970·17-s − 0.292·19-s − 0.834·23-s + 2.65·25-s + 0.422·29-s + 0.179·31-s + 0.867·37-s + 1.64·41-s − 1.10·43-s − 0.875·47-s − 0.236·53-s − 2.46·55-s + 0.816·59-s − 1.28·61-s − 0.676·65-s + 0.888·67-s − 0.237·71-s − 0.383·73-s − 0.399·79-s − 0.0301·83-s + 1.85·85-s + 0.482·89-s + 0.559·95-s − 1.65·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{3528} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good5 \( 1 + 4.27T + 5T^{2} \)
11 \( 1 - 4.27T + 11T^{2} \)
13 \( 1 - 1.27T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 1.27T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 2.27T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 - 5.27T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 7.27T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 1.72T + 53T^{2} \)
59 \( 1 - 6.27T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 7.27T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 3.27T + 73T^{2} \)
79 \( 1 + 3.54T + 79T^{2} \)
83 \( 1 + 0.274T + 83T^{2} \)
89 \( 1 - 4.54T + 89T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−8.199437375520808117057045065771, −7.56084715494105884112333032060, −6.71654288310414967634349763150, −6.23841162056677075120995915228, −4.85544098925798401277104215023, −4.12216009244573645365194056071, −3.78813135785413988787015943187, −2.70025665596136581460930474397, −1.23834149436161297774796709499, 0, 1.23834149436161297774796709499, 2.70025665596136581460930474397, 3.78813135785413988787015943187, 4.12216009244573645365194056071, 4.85544098925798401277104215023, 6.23841162056677075120995915228, 6.71654288310414967634349763150, 7.56084715494105884112333032060, 8.199437375520808117057045065771

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