Properties

Label 2-6e4-1.1-c3-0-24
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  + 1.69·5-s + 17.1·7-s + 4.00·11-s + 41.0·13-s + 3.32·17-s − 108.·19-s + 142.·23-s − 122.·25-s + 295.·29-s + 239.·31-s + 29.0·35-s − 121.·37-s − 344.·41-s + 420.·43-s + 92.7·47-s − 49.1·49-s + 191.·53-s + 6.77·55-s − 661.·59-s − 359.·61-s + 69.4·65-s + 273.·67-s + 344.·71-s − 824.·73-s + 68.6·77-s − 289.·79-s + 1.32e3·83-s + ⋯
L(s)  = 1  + 0.151·5-s + 0.925·7-s + 0.109·11-s + 0.875·13-s + 0.0474·17-s − 1.31·19-s + 1.29·23-s − 0.977·25-s + 1.89·29-s + 1.38·31-s + 0.140·35-s − 0.540·37-s − 1.31·41-s + 1.49·43-s + 0.287·47-s − 0.143·49-s + 0.495·53-s + 0.0166·55-s − 1.46·59-s − 0.754·61-s + 0.132·65-s + 0.499·67-s + 0.575·71-s − 1.32·73-s + 0.101·77-s − 0.411·79-s + 1.74·83-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: \(0\)
Selberg data: \((2,\ 1296,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.701455401\)
\(L(\frac12)\) \(\approx\) \(2.701455401\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.69T + 125T^{2} \)
7 \( 1 - 17.1T + 343T^{2} \)
11 \( 1 - 4.00T + 1.33e3T^{2} \)
13 \( 1 - 41.0T + 2.19e3T^{2} \)
17 \( 1 - 3.32T + 4.91e3T^{2} \)
19 \( 1 + 108.T + 6.85e3T^{2} \)
23 \( 1 - 142.T + 1.21e4T^{2} \)
29 \( 1 - 295.T + 2.43e4T^{2} \)
31 \( 1 - 239.T + 2.97e4T^{2} \)
37 \( 1 + 121.T + 5.06e4T^{2} \)
41 \( 1 + 344.T + 6.89e4T^{2} \)
43 \( 1 - 420.T + 7.95e4T^{2} \)
47 \( 1 - 92.7T + 1.03e5T^{2} \)
53 \( 1 - 191.T + 1.48e5T^{2} \)
59 \( 1 + 661.T + 2.05e5T^{2} \)
61 \( 1 + 359.T + 2.26e5T^{2} \)
67 \( 1 - 273.T + 3.00e5T^{2} \)
71 \( 1 - 344.T + 3.57e5T^{2} \)
73 \( 1 + 824.T + 3.89e5T^{2} \)
79 \( 1 + 289.T + 4.93e5T^{2} \)
83 \( 1 - 1.32e3T + 5.71e5T^{2} \)
89 \( 1 - 328.T + 7.04e5T^{2} \)
97 \( 1 - 767.T + 9.12e5T^{2} \)
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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

−9.083464329553399908975765217620, −8.486656837727897356989805297275, −7.82771958907433765705869405797, −6.69514698225835535913277059367, −6.05995243617628349402039296326, −4.92234726133683593408776957576, −4.28126007418816688433246451304, −3.05904046570455480886931896744, −1.89916422415378424209804763768, −0.869126288529226887965240023222, 0.869126288529226887965240023222, 1.89916422415378424209804763768, 3.05904046570455480886931896744, 4.28126007418816688433246451304, 4.92234726133683593408776957576, 6.05995243617628349402039296326, 6.69514698225835535913277059367, 7.82771958907433765705869405797, 8.486656837727897356989805297275, 9.083464329553399908975765217620

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