Properties

Label 2-6e4-1.1-c3-0-15
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  + 0.803·5-s + 2.39·7-s − 59.5·11-s + 77.7·13-s + 2.84·17-s + 118.·19-s − 110.·23-s − 124.·25-s + 125.·29-s − 63.0·31-s + 1.92·35-s − 227.·37-s + 324.·41-s + 272.·43-s + 4.93·47-s − 337.·49-s − 598.·53-s − 47.8·55-s + 670.·59-s + 464.·61-s + 62.4·65-s + 769.·67-s + 611.·71-s + 923.·73-s − 142.·77-s − 39.8·79-s − 443.·83-s + ⋯
L(s)  = 1  + 0.0718·5-s + 0.129·7-s − 1.63·11-s + 1.65·13-s + 0.0405·17-s + 1.42·19-s − 1.00·23-s − 0.994·25-s + 0.804·29-s − 0.365·31-s + 0.00928·35-s − 1.01·37-s + 1.23·41-s + 0.967·43-s + 0.0153·47-s − 0.983·49-s − 1.55·53-s − 0.117·55-s + 1.47·59-s + 0.975·61-s + 0.119·65-s + 1.40·67-s + 1.02·71-s + 1.48·73-s − 0.210·77-s − 0.0566·79-s − 0.586·83-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

\(L(2)\) \(\approx\) \(2.025272850\)
\(L(\frac12)\) \(\approx\) \(2.025272850\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.803T + 125T^{2} \)
7 \( 1 - 2.39T + 343T^{2} \)
11 \( 1 + 59.5T + 1.33e3T^{2} \)
13 \( 1 - 77.7T + 2.19e3T^{2} \)
17 \( 1 - 2.84T + 4.91e3T^{2} \)
19 \( 1 - 118.T + 6.85e3T^{2} \)
23 \( 1 + 110.T + 1.21e4T^{2} \)
29 \( 1 - 125.T + 2.43e4T^{2} \)
31 \( 1 + 63.0T + 2.97e4T^{2} \)
37 \( 1 + 227.T + 5.06e4T^{2} \)
41 \( 1 - 324.T + 6.89e4T^{2} \)
43 \( 1 - 272.T + 7.95e4T^{2} \)
47 \( 1 - 4.93T + 1.03e5T^{2} \)
53 \( 1 + 598.T + 1.48e5T^{2} \)
59 \( 1 - 670.T + 2.05e5T^{2} \)
61 \( 1 - 464.T + 2.26e5T^{2} \)
67 \( 1 - 769.T + 3.00e5T^{2} \)
71 \( 1 - 611.T + 3.57e5T^{2} \)
73 \( 1 - 923.T + 3.89e5T^{2} \)
79 \( 1 + 39.8T + 4.93e5T^{2} \)
83 \( 1 + 443.T + 5.71e5T^{2} \)
89 \( 1 + 78.4T + 7.04e5T^{2} \)
97 \( 1 + 91.0T + 9.12e5T^{2} \)
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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

−9.379672084444096027763302875749, −8.115143953180607255257059295836, −8.018373446194004118376377157147, −6.81409311296282163661734433121, −5.77535894394819124548484104254, −5.27671727657571858877309190209, −4.02516875346954731614892325670, −3.13038401750687970293511796172, −1.99245366402770298080194394048, −0.71857809037027542868175965555, 0.71857809037027542868175965555, 1.99245366402770298080194394048, 3.13038401750687970293511796172, 4.02516875346954731614892325670, 5.27671727657571858877309190209, 5.77535894394819124548484104254, 6.81409311296282163661734433121, 8.018373446194004118376377157147, 8.115143953180607255257059295836, 9.379672084444096027763302875749

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