Properties

Label 2-3528-1.1-c1-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3·11-s − 4·13-s + 4·19-s − 8·23-s − 4·25-s + 3·29-s + 5·31-s + 8·37-s + 8·41-s + 6·43-s + 10·47-s − 9·53-s + 3·55-s − 5·59-s + 10·61-s + 4·65-s + 6·67-s − 10·71-s − 2·73-s + 11·79-s + 7·83-s − 18·89-s − 4·95-s + 17·97-s − 2·101-s + 11·107-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.904·11-s − 1.10·13-s + 0.917·19-s − 1.66·23-s − 4/5·25-s + 0.557·29-s + 0.898·31-s + 1.31·37-s + 1.24·41-s + 0.914·43-s + 1.45·47-s − 1.23·53-s + 0.404·55-s − 0.650·59-s + 1.28·61-s + 0.496·65-s + 0.733·67-s − 1.18·71-s − 0.234·73-s + 1.23·79-s + 0.768·83-s − 1.90·89-s − 0.410·95-s + 1.72·97-s − 0.199·101-s + 1.06·107-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: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.251073185\)
\(L(\frac12)\) \(\approx\) \(1.251073185\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−8.360451959638380886764948992387, −7.69460985079885241062719833772, −7.43884217549794085426971367853, −6.22611532765628300070427835172, −5.60267032534603783427998213305, −4.66740916943855638629991224884, −4.03863894567815535080980917834, −2.88475291513598245872161177124, −2.20706871286241338218264018898, −0.63422137058927113139572540422, 0.63422137058927113139572540422, 2.20706871286241338218264018898, 2.88475291513598245872161177124, 4.03863894567815535080980917834, 4.66740916943855638629991224884, 5.60267032534603783427998213305, 6.22611532765628300070427835172, 7.43884217549794085426971367853, 7.69460985079885241062719833772, 8.360451959638380886764948992387

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