Properties

Label 2-3528-1.1-c1-0-36
Degree $2$
Conductor $3528$
Sign $-1$
Analytic cond. $28.1712$
Root an. cond. $5.30765$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s − 2·13-s + 3·17-s − 5·19-s + 3·23-s − 4·25-s + 6·29-s + 31-s − 5·37-s − 10·41-s − 4·43-s + 47-s + 9·53-s − 55-s + 3·59-s − 3·61-s + 2·65-s + 11·67-s − 16·71-s − 7·73-s − 11·79-s − 4·83-s − 3·85-s − 9·89-s + 5·95-s − 6·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 0.554·13-s + 0.727·17-s − 1.14·19-s + 0.625·23-s − 4/5·25-s + 1.11·29-s + 0.179·31-s − 0.821·37-s − 1.56·41-s − 0.609·43-s + 0.145·47-s + 1.23·53-s − 0.134·55-s + 0.390·59-s − 0.384·61-s + 0.248·65-s + 1.34·67-s − 1.89·71-s − 0.819·73-s − 1.23·79-s − 0.439·83-s − 0.325·85-s − 0.953·89-s + 0.512·95-s − 0.609·97-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: $-1$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
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 \( 1 + T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.431396288324148834445958757895, −7.35766335234528961236796843596, −6.85687384852181943756894044732, −5.96762258008171539796045436910, −5.10147258496194378753871243651, −4.32674865776305731068960650354, −3.51429950446952328560779724959, −2.57984956270275536749704073115, −1.43176964190127666188320781780, 0, 1.43176964190127666188320781780, 2.57984956270275536749704073115, 3.51429950446952328560779724959, 4.32674865776305731068960650354, 5.10147258496194378753871243651, 5.96762258008171539796045436910, 6.85687384852181943756894044732, 7.35766335234528961236796843596, 8.431396288324148834445958757895

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