Properties

Label 2-6e4-1.1-c3-0-40
Degree $2$
Conductor $1296$
Sign $-1$
Analytic cond. $76.4664$
Root an. cond. $8.74451$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 19.8·5-s + 5.87·7-s − 18.7·11-s + 45.8·13-s + 16.8·17-s + 10.3·19-s − 49.8·23-s + 269.·25-s + 10.9·29-s − 151.·31-s − 116.·35-s + 346.·37-s + 264.·41-s + 411.·43-s − 472.·47-s − 308.·49-s + 290.·53-s + 372.·55-s − 53.2·59-s − 293.·61-s − 911.·65-s + 398.·67-s + 647.·71-s − 478.·73-s − 110.·77-s − 374.·79-s − 933.·83-s + ⋯
L(s)  = 1  − 1.77·5-s + 0.316·7-s − 0.513·11-s + 0.978·13-s + 0.240·17-s + 0.124·19-s − 0.452·23-s + 2.15·25-s + 0.0698·29-s − 0.878·31-s − 0.563·35-s + 1.53·37-s + 1.00·41-s + 1.46·43-s − 1.46·47-s − 0.899·49-s + 0.752·53-s + 0.912·55-s − 0.117·59-s − 0.616·61-s − 1.73·65-s + 0.726·67-s + 1.08·71-s − 0.766·73-s − 0.162·77-s − 0.533·79-s − 1.23·83-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: no
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 \( 1 + 19.8T + 125T^{2} \)
7 \( 1 - 5.87T + 343T^{2} \)
11 \( 1 + 18.7T + 1.33e3T^{2} \)
13 \( 1 - 45.8T + 2.19e3T^{2} \)
17 \( 1 - 16.8T + 4.91e3T^{2} \)
19 \( 1 - 10.3T + 6.85e3T^{2} \)
23 \( 1 + 49.8T + 1.21e4T^{2} \)
29 \( 1 - 10.9T + 2.43e4T^{2} \)
31 \( 1 + 151.T + 2.97e4T^{2} \)
37 \( 1 - 346.T + 5.06e4T^{2} \)
41 \( 1 - 264.T + 6.89e4T^{2} \)
43 \( 1 - 411.T + 7.95e4T^{2} \)
47 \( 1 + 472.T + 1.03e5T^{2} \)
53 \( 1 - 290.T + 1.48e5T^{2} \)
59 \( 1 + 53.2T + 2.05e5T^{2} \)
61 \( 1 + 293.T + 2.26e5T^{2} \)
67 \( 1 - 398.T + 3.00e5T^{2} \)
71 \( 1 - 647.T + 3.57e5T^{2} \)
73 \( 1 + 478.T + 3.89e5T^{2} \)
79 \( 1 + 374.T + 4.93e5T^{2} \)
83 \( 1 + 933.T + 5.71e5T^{2} \)
89 \( 1 - 368.T + 7.04e5T^{2} \)
97 \( 1 - 274.T + 9.12e5T^{2} \)
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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

−8.649943544204584098382883444364, −7.925995574502723661052342186677, −7.57353856688698253154020831817, −6.48582211783719909102685364995, −5.42857657445758727538791160747, −4.35506580194251090520503546963, −3.77743990886932100527122152903, −2.76308468139932512643249557655, −1.14123685937038946307815989008, 0, 1.14123685937038946307815989008, 2.76308468139932512643249557655, 3.77743990886932100527122152903, 4.35506580194251090520503546963, 5.42857657445758727538791160747, 6.48582211783719909102685364995, 7.57353856688698253154020831817, 7.925995574502723661052342186677, 8.649943544204584098382883444364

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