Properties

Label 2-6e4-1.1-c3-0-16
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.3·5-s + 5.11·7-s + 55.9·11-s + 37.5·13-s − 23.6·17-s − 39.0·19-s + 71.0·23-s − 17.4·25-s + 28.3·29-s − 12.8·31-s − 53.0·35-s − 180.·37-s + 215.·41-s − 61.2·43-s − 61.8·47-s − 316.·49-s − 492.·53-s − 580.·55-s + 789.·59-s + 521.·61-s − 389.·65-s − 304.·67-s + 270.·71-s − 925.·73-s + 286.·77-s + 1.28e3·79-s + 713.·83-s + ⋯
L(s)  = 1  − 0.927·5-s + 0.276·7-s + 1.53·11-s + 0.801·13-s − 0.337·17-s − 0.471·19-s + 0.644·23-s − 0.139·25-s + 0.181·29-s − 0.0746·31-s − 0.256·35-s − 0.800·37-s + 0.820·41-s − 0.217·43-s − 0.192·47-s − 0.923·49-s − 1.27·53-s − 1.42·55-s + 1.74·59-s + 1.09·61-s − 0.743·65-s − 0.555·67-s + 0.451·71-s − 1.48·73-s + 0.423·77-s + 1.83·79-s + 0.944·83-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: $\chi_{1296} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.957726263\)
\(L(\frac12)\) \(\approx\) \(1.957726263\)
\(L(\frac{5}{2})\) 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 \( 1 + 10.3T + 125T^{2} \)
7 \( 1 - 5.11T + 343T^{2} \)
11 \( 1 - 55.9T + 1.33e3T^{2} \)
13 \( 1 - 37.5T + 2.19e3T^{2} \)
17 \( 1 + 23.6T + 4.91e3T^{2} \)
19 \( 1 + 39.0T + 6.85e3T^{2} \)
23 \( 1 - 71.0T + 1.21e4T^{2} \)
29 \( 1 - 28.3T + 2.43e4T^{2} \)
31 \( 1 + 12.8T + 2.97e4T^{2} \)
37 \( 1 + 180.T + 5.06e4T^{2} \)
41 \( 1 - 215.T + 6.89e4T^{2} \)
43 \( 1 + 61.2T + 7.95e4T^{2} \)
47 \( 1 + 61.8T + 1.03e5T^{2} \)
53 \( 1 + 492.T + 1.48e5T^{2} \)
59 \( 1 - 789.T + 2.05e5T^{2} \)
61 \( 1 - 521.T + 2.26e5T^{2} \)
67 \( 1 + 304.T + 3.00e5T^{2} \)
71 \( 1 - 270.T + 3.57e5T^{2} \)
73 \( 1 + 925.T + 3.89e5T^{2} \)
79 \( 1 - 1.28e3T + 4.93e5T^{2} \)
83 \( 1 - 713.T + 5.71e5T^{2} \)
89 \( 1 - 404.T + 7.04e5T^{2} \)
97 \( 1 - 75.0T + 9.12e5T^{2} \)
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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

−9.055412383583005801301726148390, −8.566084606082056203575166443185, −7.69649474140189380710520573023, −6.78886962675928966258991808274, −6.14756749502821721631074983438, −4.87079450314752286688131428849, −4.01200807030069287981224628238, −3.39093446666749378375077464387, −1.84358428810393171001652714222, −0.72948687510697993171544515695, 0.72948687510697993171544515695, 1.84358428810393171001652714222, 3.39093446666749378375077464387, 4.01200807030069287981224628238, 4.87079450314752286688131428849, 6.14756749502821721631074983438, 6.78886962675928966258991808274, 7.69649474140189380710520573023, 8.566084606082056203575166443185, 9.055412383583005801301726148390

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