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Properties

Label 2-3528-1.1-c1-0-29
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  − 4·5-s + 3·13-s + 4·17-s − 7·19-s + 4·23-s + 11·25-s − 8·29-s + 5·31-s + 3·37-s − 8·41-s + 11·43-s − 4·47-s + 4·53-s − 12·59-s + 2·61-s − 12·65-s − 3·67-s + 12·71-s − 73-s + 79-s − 12·83-s − 16·85-s − 8·89-s + 28·95-s + 2·97-s − 3·103-s − 12·107-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.832·13-s + 0.970·17-s − 1.60·19-s + 0.834·23-s + 11/5·25-s − 1.48·29-s + 0.898·31-s + 0.493·37-s − 1.24·41-s + 1.67·43-s − 0.583·47-s + 0.549·53-s − 1.56·59-s + 0.256·61-s − 1.48·65-s − 0.366·67-s + 1.42·71-s − 0.117·73-s + 0.112·79-s − 1.31·83-s − 1.73·85-s − 0.847·89-s + 2.87·95-s + 0.203·97-s − 0.295·103-s − 1.16·107-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: Trivial
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.194361501970396886120152797440, −7.57418679373738242276396502503, −6.85456593432273407333510212015, −6.03280200199091137882695946672, −5.00545424525312625017162398567, −4.12876132755004627839190108364, −3.67505440632787502402844253708, −2.74640902963781918921748663206, −1.23402938330046163739041463301, 0, 1.23402938330046163739041463301, 2.74640902963781918921748663206, 3.67505440632787502402844253708, 4.12876132755004627839190108364, 5.00545424525312625017162398567, 6.03280200199091137882695946672, 6.85456593432273407333510212015, 7.57418679373738242276396502503, 8.194361501970396886120152797440

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