Properties

Label 2-6864-1.1-c1-0-41
Degree $2$
Conductor $6864$
Sign $1$
Analytic cond. $54.8093$
Root an. cond. $7.40333$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 1.39·5-s + 3.28·7-s + 9-s + 11-s − 13-s − 1.39·15-s − 2.67·17-s − 3.28·19-s + 3.28·21-s + 3.28·23-s − 3.06·25-s + 27-s + 1.89·29-s − 3.39·31-s + 33-s − 4.56·35-s + 4·37-s − 39-s + 5.28·41-s + 1.89·43-s − 1.39·45-s + 7.34·47-s + 3.78·49-s − 2.67·51-s + 11.3·53-s − 1.39·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.622·5-s + 1.24·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s − 0.359·15-s − 0.648·17-s − 0.753·19-s + 0.716·21-s + 0.684·23-s − 0.613·25-s + 0.192·27-s + 0.351·29-s − 0.609·31-s + 0.174·33-s − 0.771·35-s + 0.657·37-s − 0.160·39-s + 0.825·41-s + 0.288·43-s − 0.207·45-s + 1.07·47-s + 0.540·49-s − 0.374·51-s + 1.55·53-s − 0.187·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6864\)    =    \(2^{4} \cdot 3 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(54.8093\)
Root analytic conductor: \(7.40333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.573278101\)
\(L(\frac12)\) \(\approx\) \(2.573278101\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
good5 \( 1 + 1.39T + 5T^{2} \)
7 \( 1 - 3.28T + 7T^{2} \)
17 \( 1 + 2.67T + 17T^{2} \)
19 \( 1 + 3.28T + 19T^{2} \)
23 \( 1 - 3.28T + 23T^{2} \)
29 \( 1 - 1.89T + 29T^{2} \)
31 \( 1 + 3.39T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 5.28T + 41T^{2} \)
43 \( 1 - 1.89T + 43T^{2} \)
47 \( 1 - 7.34T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 1.21T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 - 3.17T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 7.49T + 73T^{2} \)
79 \( 1 + 3.32T + 79T^{2} \)
83 \( 1 - 7.78T + 83T^{2} \)
89 \( 1 - 6.17T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−7.928182949313093825972165831626, −7.44870107752622900495991665136, −6.77312033113064948809160245417, −5.80986490165890716962074289978, −4.94175705112604354213097585156, −4.26307987328130835787419995366, −3.78498142240313217101110121048, −2.58267506080754210229162123079, −1.95402874237965076822697188198, −0.808163852510678665345519168961, 0.808163852510678665345519168961, 1.95402874237965076822697188198, 2.58267506080754210229162123079, 3.78498142240313217101110121048, 4.26307987328130835787419995366, 4.94175705112604354213097585156, 5.80986490165890716962074289978, 6.77312033113064948809160245417, 7.44870107752622900495991665136, 7.928182949313093825972165831626

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