Properties

Label 2-6864-1.1-c1-0-95
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 − 0.585·5-s + 2.82·7-s + 9-s + 11-s − 13-s + 0.585·15-s + 2.24·17-s − 2.82·21-s − 0.828·23-s − 4.65·25-s − 27-s − 3.41·29-s + 2.58·31-s − 33-s − 1.65·35-s − 4.82·37-s + 39-s − 7.65·41-s − 8.24·43-s − 0.585·45-s + 8·47-s + 1.00·49-s − 2.24·51-s − 13.3·53-s − 0.585·55-s + 2.82·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.261·5-s + 1.06·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s + 0.151·15-s + 0.543·17-s − 0.617·21-s − 0.172·23-s − 0.931·25-s − 0.192·27-s − 0.634·29-s + 0.464·31-s − 0.174·33-s − 0.280·35-s − 0.793·37-s + 0.160·39-s − 1.19·41-s − 1.25·43-s − 0.0873·45-s + 1.16·47-s + 0.142·49-s − 0.314·51-s − 1.82·53-s − 0.0789·55-s + 0.368·59-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: \(1\)
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 \( 1 + 0.585T + 5T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 0.828T + 23T^{2} \)
29 \( 1 + 3.41T + 29T^{2} \)
31 \( 1 - 2.58T + 31T^{2} \)
37 \( 1 + 4.82T + 37T^{2} \)
41 \( 1 + 7.65T + 41T^{2} \)
43 \( 1 + 8.24T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + 8.82T + 61T^{2} \)
67 \( 1 - 4.24T + 67T^{2} \)
71 \( 1 - 5.17T + 71T^{2} \)
73 \( 1 - 5.65T + 73T^{2} \)
79 \( 1 + 7.07T + 79T^{2} \)
83 \( 1 - 6.34T + 83T^{2} \)
89 \( 1 - 7.89T + 89T^{2} \)
97 \( 1 + 7.17T + 97T^{2} \)
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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

−7.74883021252135974222160138896, −6.90209548429201878135619780355, −6.21273687584574642043414012428, −5.33834051989332735929831736486, −4.90325343681183329991013133959, −4.06802266553851319487250904221, −3.31451382651844777120285372680, −2.03980195635201185504789478817, −1.33326412629552105195310475124, 0, 1.33326412629552105195310475124, 2.03980195635201185504789478817, 3.31451382651844777120285372680, 4.06802266553851319487250904221, 4.90325343681183329991013133959, 5.33834051989332735929831736486, 6.21273687584574642043414012428, 6.90209548429201878135619780355, 7.74883021252135974222160138896

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