Properties

Label 2-6e4-1.1-c1-0-4
Degree $2$
Conductor $1296$
Sign $1$
Analytic cond. $10.3486$
Root an. cond. $3.21692$
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  − 2·7-s + 3·11-s + 2·13-s − 3·17-s + 19-s + 6·23-s − 5·25-s + 6·29-s + 4·31-s − 4·37-s + 9·41-s + 43-s + 6·47-s − 3·49-s + 12·53-s − 3·59-s + 8·61-s − 5·67-s + 12·71-s + 11·73-s − 6·77-s + 4·79-s − 12·83-s + 6·89-s − 4·91-s + 5·97-s − 14·103-s + ⋯
L(s)  = 1  − 0.755·7-s + 0.904·11-s + 0.554·13-s − 0.727·17-s + 0.229·19-s + 1.25·23-s − 25-s + 1.11·29-s + 0.718·31-s − 0.657·37-s + 1.40·41-s + 0.152·43-s + 0.875·47-s − 3/7·49-s + 1.64·53-s − 0.390·59-s + 1.02·61-s − 0.610·67-s + 1.42·71-s + 1.28·73-s − 0.683·77-s + 0.450·79-s − 1.31·83-s + 0.635·89-s − 0.419·91-s + 0.507·97-s − 1.37·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.616972048\)
\(L(\frac12)\) \(\approx\) \(1.616972048\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−9.515144727195132739534503821127, −8.982636192572195812922115743561, −8.131333319129826356153375895616, −6.97337389606279696915121240099, −6.48609808834188223043361623015, −5.57014449737897797729246275255, −4.39002474802971059462229776672, −3.57376731663222433353058317817, −2.48468545989121014995168597575, −0.967496248641159004586773303229, 0.967496248641159004586773303229, 2.48468545989121014995168597575, 3.57376731663222433353058317817, 4.39002474802971059462229776672, 5.57014449737897797729246275255, 6.48609808834188223043361623015, 6.97337389606279696915121240099, 8.131333319129826356153375895616, 8.982636192572195812922115743561, 9.515144727195132739534503821127

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