Properties

Label 2-6864-1.1-c1-0-118
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 + 3.97·5-s − 3.17·7-s + 9-s + 11-s + 13-s + 3.97·15-s − 7.80·17-s − 7.17·19-s − 3.17·21-s − 3.36·23-s + 10.7·25-s + 27-s − 7.61·29-s − 3.15·31-s + 33-s − 12.6·35-s + 2.93·37-s + 39-s + 1.64·41-s + 4.15·43-s + 3.97·45-s − 0.660·47-s + 3.06·49-s − 7.80·51-s + 0.0696·53-s + 3.97·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.77·5-s − 1.19·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 1.02·15-s − 1.89·17-s − 1.64·19-s − 0.692·21-s − 0.700·23-s + 2.15·25-s + 0.192·27-s − 1.41·29-s − 0.566·31-s + 0.174·33-s − 2.13·35-s + 0.482·37-s + 0.160·39-s + 0.256·41-s + 0.633·43-s + 0.592·45-s − 0.0963·47-s + 0.438·49-s − 1.09·51-s + 0.00957·53-s + 0.535·55-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 \( 1 - 3.97T + 5T^{2} \)
7 \( 1 + 3.17T + 7T^{2} \)
17 \( 1 + 7.80T + 17T^{2} \)
19 \( 1 + 7.17T + 19T^{2} \)
23 \( 1 + 3.36T + 23T^{2} \)
29 \( 1 + 7.61T + 29T^{2} \)
31 \( 1 + 3.15T + 31T^{2} \)
37 \( 1 - 2.93T + 37T^{2} \)
41 \( 1 - 1.64T + 41T^{2} \)
43 \( 1 - 4.15T + 43T^{2} \)
47 \( 1 + 0.660T + 47T^{2} \)
53 \( 1 - 0.0696T + 53T^{2} \)
59 \( 1 + 8.88T + 59T^{2} \)
61 \( 1 + 5.47T + 61T^{2} \)
67 \( 1 + 5.50T + 67T^{2} \)
71 \( 1 + 2.11T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 - 0.965T + 89T^{2} \)
97 \( 1 - 7.31T + 97T^{2} \)
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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

−7.49761772556142563741979952646, −6.69284306224787905548586320325, −6.16674892603518692111932810955, −5.88783007870307444912047054212, −4.63721565248165518575451355766, −3.98113670771878157020141866996, −2.95689695481874564368345763472, −2.19902167978871740264068120387, −1.71586815524691770531477982276, 0, 1.71586815524691770531477982276, 2.19902167978871740264068120387, 2.95689695481874564368345763472, 3.98113670771878157020141866996, 4.63721565248165518575451355766, 5.88783007870307444912047054212, 6.16674892603518692111932810955, 6.69284306224787905548586320325, 7.49761772556142563741979952646

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