Properties

Label 2-6e4-1.1-c1-0-8
Degree $2$
Conductor $1296$
Sign $1$
Analytic cond. $10.3486$
Root an. cond. $3.21692$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 7-s − 3·11-s − 13-s + 6·17-s + 4·19-s + 3·23-s + 4·25-s + 3·29-s − 5·31-s + 3·35-s + 2·37-s + 3·41-s + 43-s + 9·47-s − 6·49-s − 6·53-s − 9·55-s + 3·59-s − 13·61-s − 3·65-s + 7·67-s + 12·71-s − 10·73-s − 3·77-s − 11·79-s + 9·83-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s − 0.904·11-s − 0.277·13-s + 1.45·17-s + 0.917·19-s + 0.625·23-s + 4/5·25-s + 0.557·29-s − 0.898·31-s + 0.507·35-s + 0.328·37-s + 0.468·41-s + 0.152·43-s + 1.31·47-s − 6/7·49-s − 0.824·53-s − 1.21·55-s + 0.390·59-s − 1.66·61-s − 0.372·65-s + 0.855·67-s + 1.42·71-s − 1.17·73-s − 0.341·77-s − 1.23·79-s + 0.987·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1296} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.230374188\)
\(L(\frac12)\) \(\approx\) \(2.230374188\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−9.702090382872232304706168176817, −9.044324062031316603425575486661, −7.910252938778095276305610373978, −7.33863505766548836719037800126, −6.13181657170124013100903903614, −5.45456509339651679693427315165, −4.86009350217895805294255984450, −3.31098434535709278954863783943, −2.37448751408578393336540536078, −1.21023528090684925585240090874, 1.21023528090684925585240090874, 2.37448751408578393336540536078, 3.31098434535709278954863783943, 4.86009350217895805294255984450, 5.45456509339651679693427315165, 6.13181657170124013100903903614, 7.33863505766548836719037800126, 7.910252938778095276305610373978, 9.044324062031316603425575486661, 9.702090382872232304706168176817

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