Properties

Label 2-3528-1.1-c1-0-50
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  + 2·5-s + 4·11-s − 2·13-s − 6·17-s − 8·19-s − 25-s − 6·29-s − 8·31-s − 2·37-s + 2·41-s − 4·43-s − 8·47-s − 6·53-s + 8·55-s + 6·61-s − 4·65-s − 4·67-s + 8·71-s − 10·73-s + 16·79-s + 8·83-s − 12·85-s − 6·89-s − 16·95-s + 6·97-s + 2·101-s + 16·103-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.20·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 1.16·47-s − 0.824·53-s + 1.07·55-s + 0.768·61-s − 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 1.80·79-s + 0.878·83-s − 1.30·85-s − 0.635·89-s − 1.64·95-s + 0.609·97-s + 0.199·101-s + 1.57·103-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−8.373309408931680629637484917349, −7.32119224555753352111842303305, −6.52917361486339326450126089747, −6.17908275353188198229198176353, −5.16177791023533130836977920691, −4.33405029272495229169198202320, −3.58097051005902557419651238987, −2.16319834122457639545883386464, −1.80766814068560455754339816128, 0, 1.80766814068560455754339816128, 2.16319834122457639545883386464, 3.58097051005902557419651238987, 4.33405029272495229169198202320, 5.16177791023533130836977920691, 6.17908275353188198229198176353, 6.52917361486339326450126089747, 7.32119224555753352111842303305, 8.373309408931680629637484917349

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