Properties

Label 2-6e4-1.1-c1-0-10
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 − 2·7-s + 6·11-s + 5·13-s − 3·17-s − 2·19-s − 6·23-s + 4·25-s + 3·29-s + 4·31-s − 6·35-s + 5·37-s − 6·41-s + 10·43-s − 3·49-s − 6·53-s + 18·55-s + 12·59-s + 5·61-s + 15·65-s − 2·67-s − 6·71-s − 73-s − 12·77-s + 10·79-s − 9·85-s − 3·89-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.755·7-s + 1.80·11-s + 1.38·13-s − 0.727·17-s − 0.458·19-s − 1.25·23-s + 4/5·25-s + 0.557·29-s + 0.718·31-s − 1.01·35-s + 0.821·37-s − 0.937·41-s + 1.52·43-s − 3/7·49-s − 0.824·53-s + 2.42·55-s + 1.56·59-s + 0.640·61-s + 1.86·65-s − 0.244·67-s − 0.712·71-s − 0.117·73-s − 1.36·77-s + 1.12·79-s − 0.976·85-s − 0.317·89-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.223231888\)
\(L(\frac12)\) \(\approx\) \(2.223231888\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−9.604404195328056888635195624166, −9.007248355470093411359381448674, −8.298117991686099747490643532078, −6.76359618838495100744614032144, −6.29860250743506028393247154678, −5.87100059007609562623927091882, −4.38712587564334162926568406315, −3.59193545168410033442089288096, −2.26841618983682361893117043506, −1.22349289581102954381038244904, 1.22349289581102954381038244904, 2.26841618983682361893117043506, 3.59193545168410033442089288096, 4.38712587564334162926568406315, 5.87100059007609562623927091882, 6.29860250743506028393247154678, 6.76359618838495100744614032144, 8.298117991686099747490643532078, 9.007248355470093411359381448674, 9.604404195328056888635195624166

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