Properties

Label 2-6864-1.1-c1-0-49
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 + 1.74·5-s + 4.53·7-s + 9-s − 11-s + 13-s − 1.74·15-s − 1.63·17-s − 0.534·19-s − 4.53·21-s + 7.60·23-s − 1.95·25-s − 27-s − 1.21·29-s + 2.67·31-s + 33-s + 7.91·35-s − 9.91·37-s − 39-s + 3.37·41-s − 3.63·43-s + 1.74·45-s + 9.48·47-s + 13.5·49-s + 1.63·51-s − 3.57·53-s − 1.74·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.780·5-s + 1.71·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s − 0.450·15-s − 0.395·17-s − 0.122·19-s − 0.989·21-s + 1.58·23-s − 0.391·25-s − 0.192·27-s − 0.224·29-s + 0.480·31-s + 0.174·33-s + 1.33·35-s − 1.62·37-s − 0.160·39-s + 0.527·41-s − 0.553·43-s + 0.260·45-s + 1.38·47-s + 1.93·49-s + 0.228·51-s − 0.491·53-s − 0.235·55-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: $\chi_{6864} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.550455187\)
\(L(\frac12)\) \(\approx\) \(2.550455187\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.74T + 5T^{2} \)
7 \( 1 - 4.53T + 7T^{2} \)
17 \( 1 + 1.63T + 17T^{2} \)
19 \( 1 + 0.534T + 19T^{2} \)
23 \( 1 - 7.60T + 23T^{2} \)
29 \( 1 + 1.21T + 29T^{2} \)
31 \( 1 - 2.67T + 31T^{2} \)
37 \( 1 + 9.91T + 37T^{2} \)
41 \( 1 - 3.37T + 41T^{2} \)
43 \( 1 + 3.63T + 43T^{2} \)
47 \( 1 - 9.48T + 47T^{2} \)
53 \( 1 + 3.57T + 53T^{2} \)
59 \( 1 - 2.64T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 3.06T + 71T^{2} \)
73 \( 1 - 4.95T + 73T^{2} \)
79 \( 1 - 1.72T + 79T^{2} \)
83 \( 1 + 1.35T + 83T^{2} \)
89 \( 1 - 6.39T + 89T^{2} \)
97 \( 1 + 2.93T + 97T^{2} \)
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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

−7.978026712640444263871169994589, −7.19791480038701363399291175378, −6.58454085803582791423679186860, −5.61567897880384939575105516275, −5.21598595998499094436677325749, −4.63534042877687707171738311504, −3.72507466828502982881287949825, −2.44483609880182626691745167249, −1.76379012993947609705281228783, −0.891932050236332853054426865322, 0.891932050236332853054426865322, 1.76379012993947609705281228783, 2.44483609880182626691745167249, 3.72507466828502982881287949825, 4.63534042877687707171738311504, 5.21598595998499094436677325749, 5.61567897880384939575105516275, 6.58454085803582791423679186860, 7.19791480038701363399291175378, 7.978026712640444263871169994589

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