Properties

Label 2-6864-1.1-c1-0-47
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 − 3.41·5-s − 2.82·7-s + 9-s + 11-s − 13-s + 3.41·15-s − 6.24·17-s + 2.82·21-s + 4.82·23-s + 6.65·25-s − 27-s − 0.585·29-s + 5.41·31-s − 33-s + 9.65·35-s + 0.828·37-s + 39-s + 3.65·41-s + 0.242·43-s − 3.41·45-s + 8·47-s + 1.00·49-s + 6.24·51-s + 9.31·53-s − 3.41·55-s − 2.82·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.52·5-s − 1.06·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s + 0.881·15-s − 1.51·17-s + 0.617·21-s + 1.00·23-s + 1.33·25-s − 0.192·27-s − 0.108·29-s + 0.972·31-s − 0.174·33-s + 1.63·35-s + 0.136·37-s + 0.160·39-s + 0.571·41-s + 0.0370·43-s − 0.508·45-s + 1.16·47-s + 0.142·49-s + 0.874·51-s + 1.27·53-s − 0.460·55-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 \( 1 + 3.41T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
17 \( 1 + 6.24T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 0.585T + 29T^{2} \)
31 \( 1 - 5.41T + 31T^{2} \)
37 \( 1 - 0.828T + 37T^{2} \)
41 \( 1 - 3.65T + 41T^{2} \)
43 \( 1 - 0.242T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 9.31T + 53T^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 3.17T + 61T^{2} \)
67 \( 1 + 4.24T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 5.65T + 73T^{2} \)
79 \( 1 - 7.07T + 79T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−7.47182294900589047155541576297, −6.83696367232453877081604308674, −6.47669262172993251323946996524, −5.48660584952677296108645475836, −4.52022094695823631568998793859, −4.12684405277911733872895397368, −3.28300473646686573795545199187, −2.45778113200640869140963331478, −0.869785714574265387661658123978, 0, 0.869785714574265387661658123978, 2.45778113200640869140963331478, 3.28300473646686573795545199187, 4.12684405277911733872895397368, 4.52022094695823631568998793859, 5.48660584952677296108645475836, 6.47669262172993251323946996524, 6.83696367232453877081604308674, 7.47182294900589047155541576297

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