Properties

Label 2-4536-1.1-c1-0-28
Degree $2$
Conductor $4536$
Sign $1$
Analytic cond. $36.2201$
Root an. cond. $6.01831$
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  + 2·5-s + 7-s + 3·11-s − 2·17-s − 2·19-s − 4·23-s − 25-s + 2·29-s + 10·31-s + 2·35-s + 9·37-s − 2·41-s − 43-s + 8·47-s + 49-s + 11·53-s + 6·55-s + 6·59-s − 8·61-s − 3·67-s + 3·71-s + 4·73-s + 3·77-s + 79-s + 6·83-s − 4·85-s + 4·89-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s + 0.904·11-s − 0.485·17-s − 0.458·19-s − 0.834·23-s − 1/5·25-s + 0.371·29-s + 1.79·31-s + 0.338·35-s + 1.47·37-s − 0.312·41-s − 0.152·43-s + 1.16·47-s + 1/7·49-s + 1.51·53-s + 0.809·55-s + 0.781·59-s − 1.02·61-s − 0.366·67-s + 0.356·71-s + 0.468·73-s + 0.341·77-s + 0.112·79-s + 0.658·83-s − 0.433·85-s + 0.423·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4536\)    =    \(2^{3} \cdot 3^{4} \cdot 7\)
Sign: $1$
Analytic conductor: \(36.2201\)
Root analytic conductor: \(6.01831\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4536,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.606204314\)
\(L(\frac12)\) \(\approx\) \(2.606204314\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.402773602693818822026247229596, −7.67188577237697185267550322771, −6.65570018730583887555357322639, −6.22662054902984970122706181668, −5.50906955083684714686429525487, −4.50449775435939553268652534010, −3.97554023181141409715836466376, −2.67970398699750458834376931624, −1.97851182872562845647314172121, −0.938284903546578701335889112674, 0.938284903546578701335889112674, 1.97851182872562845647314172121, 2.67970398699750458834376931624, 3.97554023181141409715836466376, 4.50449775435939553268652534010, 5.50906955083684714686429525487, 6.22662054902984970122706181668, 6.65570018730583887555357322639, 7.67188577237697185267550322771, 8.402773602693818822026247229596

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