Properties

Label 2-6e4-1.1-c3-0-33
Degree $2$
Conductor $1296$
Sign $-1$
Analytic cond. $76.4664$
Root an. cond. $8.74451$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 21·5-s − 8·7-s + 36·11-s − 49·13-s − 21·17-s + 112·19-s + 180·23-s + 316·25-s + 135·29-s − 308·31-s + 168·35-s − 37-s + 42·41-s − 20·43-s + 84·47-s − 279·49-s + 174·53-s − 756·55-s + 504·59-s − 385·61-s + 1.02e3·65-s − 272·67-s − 888·71-s + 371·73-s − 288·77-s + 652·79-s + 84·83-s + ⋯
L(s)  = 1  − 1.87·5-s − 0.431·7-s + 0.986·11-s − 1.04·13-s − 0.299·17-s + 1.35·19-s + 1.63·23-s + 2.52·25-s + 0.864·29-s − 1.78·31-s + 0.811·35-s − 0.00444·37-s + 0.159·41-s − 0.0709·43-s + 0.260·47-s − 0.813·49-s + 0.450·53-s − 1.85·55-s + 1.11·59-s − 0.808·61-s + 1.96·65-s − 0.495·67-s − 1.48·71-s + 0.594·73-s − 0.426·77-s + 0.928·79-s + 0.111·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $-1$
Analytic conductor: \(76.4664\)
Root analytic conductor: \(8.74451\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1296,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 21 T + p^{3} T^{2} \)
7 \( 1 + 8 T + p^{3} T^{2} \)
11 \( 1 - 36 T + p^{3} T^{2} \)
13 \( 1 + 49 T + p^{3} T^{2} \)
17 \( 1 + 21 T + p^{3} T^{2} \)
19 \( 1 - 112 T + p^{3} T^{2} \)
23 \( 1 - 180 T + p^{3} T^{2} \)
29 \( 1 - 135 T + p^{3} T^{2} \)
31 \( 1 + 308 T + p^{3} T^{2} \)
37 \( 1 + T + p^{3} T^{2} \)
41 \( 1 - 42 T + p^{3} T^{2} \)
43 \( 1 + 20 T + p^{3} T^{2} \)
47 \( 1 - 84 T + p^{3} T^{2} \)
53 \( 1 - 174 T + p^{3} T^{2} \)
59 \( 1 - 504 T + p^{3} T^{2} \)
61 \( 1 + 385 T + p^{3} T^{2} \)
67 \( 1 + 272 T + p^{3} T^{2} \)
71 \( 1 + 888 T + p^{3} T^{2} \)
73 \( 1 - 371 T + p^{3} T^{2} \)
79 \( 1 - 652 T + p^{3} T^{2} \)
83 \( 1 - 84 T + p^{3} T^{2} \)
89 \( 1 + 21 T + p^{3} T^{2} \)
97 \( 1 + 1246 T + p^{3} T^{2} \)
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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

−8.943789459903209121383830188153, −7.940723245377987317077690617612, −7.17485784174274370950965843414, −6.81991635983329139306063307616, −5.30325988333506715807666894305, −4.47697502960724786319707485114, −3.60126964613226170557300052100, −2.90204035794661186908030892407, −1.07567772053420290956943714256, 0, 1.07567772053420290956943714256, 2.90204035794661186908030892407, 3.60126964613226170557300052100, 4.47697502960724786319707485114, 5.30325988333506715807666894305, 6.81991635983329139306063307616, 7.17485784174274370950965843414, 7.940723245377987317077690617612, 8.943789459903209121383830188153

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