Properties

Label 2-4536-1.1-c1-0-15
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  − 5-s + 7-s − 6·11-s + 6·13-s − 2·17-s + 7·19-s − 23-s − 4·25-s + 2·29-s + 10·31-s − 35-s − 6·37-s − 8·41-s − 10·43-s + 8·47-s + 49-s + 2·53-s + 6·55-s + 7·61-s − 6·65-s − 12·67-s + 15·71-s − 2·73-s − 6·77-s + 79-s + 12·83-s + 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 1.80·11-s + 1.66·13-s − 0.485·17-s + 1.60·19-s − 0.208·23-s − 4/5·25-s + 0.371·29-s + 1.79·31-s − 0.169·35-s − 0.986·37-s − 1.24·41-s − 1.52·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s + 0.809·55-s + 0.896·61-s − 0.744·65-s − 1.46·67-s + 1.78·71-s − 0.234·73-s − 0.683·77-s + 0.112·79-s + 1.31·83-s + 0.216·85-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

\(L(1)\) \(\approx\) \(1.661350015\)
\(L(\frac12)\) \(\approx\) \(1.661350015\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−8.258534877886123881516881958726, −7.79067111123171966699020607366, −6.96995072676846735892694754372, −6.08177989229326585475743736013, −5.31878813596332244967742727497, −4.72715111248749756023836609092, −3.66159139267585105415203405545, −3.02811103455129536867830610709, −1.92802436486972449904496470791, −0.72081180776643933088668872598, 0.72081180776643933088668872598, 1.92802436486972449904496470791, 3.02811103455129536867830610709, 3.66159139267585105415203405545, 4.72715111248749756023836609092, 5.31878813596332244967742727497, 6.08177989229326585475743736013, 6.96995072676846735892694754372, 7.79067111123171966699020607366, 8.258534877886123881516881958726

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