Properties

Label 2-6864-1.1-c1-0-62
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.70·5-s + 1.70·7-s + 9-s + 11-s − 13-s + 3.70·15-s + 4·17-s − 6·19-s − 1.70·21-s + 0.298·23-s + 8.70·25-s − 27-s − 0.298·29-s − 5.40·31-s − 33-s − 6.29·35-s + 7.40·37-s + 39-s − 0.298·41-s − 9.10·43-s − 3.70·45-s − 4·47-s − 4.10·49-s − 4·51-s + 7.40·53-s − 3.70·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.65·5-s + 0.643·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s + 0.955·15-s + 0.970·17-s − 1.37·19-s − 0.371·21-s + 0.0622·23-s + 1.74·25-s − 0.192·27-s − 0.0554·29-s − 0.970·31-s − 0.174·33-s − 1.06·35-s + 1.21·37-s + 0.160·39-s − 0.0466·41-s − 1.38·43-s − 0.551·45-s − 0.583·47-s − 0.586·49-s − 0.560·51-s + 1.01·53-s − 0.499·55-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: \(1\)
Selberg data: \((2,\ 6864,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.70T + 5T^{2} \)
7 \( 1 - 1.70T + 7T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - 0.298T + 23T^{2} \)
29 \( 1 + 0.298T + 29T^{2} \)
31 \( 1 + 5.40T + 31T^{2} \)
37 \( 1 - 7.40T + 37T^{2} \)
41 \( 1 + 0.298T + 41T^{2} \)
43 \( 1 + 9.10T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 7.40T + 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 - 4.29T + 61T^{2} \)
67 \( 1 - 5.10T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 1.70T + 73T^{2} \)
79 \( 1 - 3.40T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 - 1.40T + 89T^{2} \)
97 \( 1 - 8.80T + 97T^{2} \)
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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

−7.61001819386563567709428677532, −7.02969208184865406969157776201, −6.31277222700592125955929369751, −5.32120795772886066610674338942, −4.72818415603061453462062339144, −3.98431247534232072523416869895, −3.48406217765785004410411241504, −2.22459892140817555067716360000, −1.04097139636600713320524204073, 0, 1.04097139636600713320524204073, 2.22459892140817555067716360000, 3.48406217765785004410411241504, 3.98431247534232072523416869895, 4.72818415603061453462062339144, 5.32120795772886066610674338942, 6.31277222700592125955929369751, 7.02969208184865406969157776201, 7.61001819386563567709428677532

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