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  + 3.27·5-s − 3.27·11-s − 6.27·13-s − 4·17-s + 6.27·19-s − 4·23-s + 5.72·25-s − 5.27·29-s + 31-s − 2.27·37-s − 4.54·41-s + 0.274·43-s − 6·47-s − 9.27·53-s − 10.7·55-s − 1.27·59-s − 10·61-s − 20.5·65-s − 0.274·67-s − 2·71-s + 4.27·73-s + 11.5·79-s + 7.27·83-s − 13.0·85-s − 10.5·89-s + 20.5·95-s − 8.72·97-s + ⋯
L(s)  = 1  + 1.46·5-s − 0.987·11-s − 1.74·13-s − 0.970·17-s + 1.43·19-s − 0.834·23-s + 1.14·25-s − 0.979·29-s + 0.179·31-s − 0.373·37-s − 0.710·41-s + 0.0419·43-s − 0.875·47-s − 1.27·53-s − 1.44·55-s − 0.165·59-s − 1.28·61-s − 2.54·65-s − 0.0335·67-s − 0.237·71-s + 0.500·73-s + 1.29·79-s + 0.798·83-s − 1.42·85-s − 1.11·89-s + 2.10·95-s − 0.885·97-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 \( 1 - 3.27T + 5T^{2} \)
11 \( 1 + 3.27T + 11T^{2} \)
13 \( 1 + 6.27T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 6.27T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 5.27T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 2.27T + 37T^{2} \)
41 \( 1 + 4.54T + 41T^{2} \)
43 \( 1 - 0.274T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 9.27T + 53T^{2} \)
59 \( 1 + 1.27T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 0.274T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 4.27T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 - 7.27T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + 8.72T + 97T^{2} \)
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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

−8.106745005014989322563050040664, −7.45301541763017165200606598950, −6.70073632245401744492253588366, −5.83356918575981281550840563833, −5.18995316409897141238366334982, −4.67468365657476875553358733735, −3.21602223139018640327563799336, −2.38853213020453115209962611679, −1.74294431929387312457142758629, 0, 1.74294431929387312457142758629, 2.38853213020453115209962611679, 3.21602223139018640327563799336, 4.67468365657476875553358733735, 5.18995316409897141238366334982, 5.83356918575981281550840563833, 6.70073632245401744492253588366, 7.45301541763017165200606598950, 8.106745005014989322563050040664

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