Properties

Label 2-6864-1.1-c1-0-5
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 − 3.18·5-s − 1.10·7-s + 9-s − 11-s − 13-s − 3.18·15-s − 6.29·17-s − 3.74·19-s − 1.10·21-s − 3.26·23-s + 5.14·25-s + 27-s + 3.60·29-s + 0.336·31-s − 33-s + 3.52·35-s + 6.37·37-s − 39-s + 0.894·41-s − 8.85·43-s − 3.18·45-s + 8.21·47-s − 5.77·49-s − 6.29·51-s + 5.32·53-s + 3.18·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.42·5-s − 0.417·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s − 0.822·15-s − 1.52·17-s − 0.858·19-s − 0.241·21-s − 0.681·23-s + 1.02·25-s + 0.192·27-s + 0.668·29-s + 0.0604·31-s − 0.174·33-s + 0.595·35-s + 1.04·37-s − 0.160·39-s + 0.139·41-s − 1.35·43-s − 0.474·45-s + 1.19·47-s − 0.825·49-s − 0.880·51-s + 0.731·53-s + 0.429·55-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

\(L(1)\) \(\approx\) \(0.9097681147\)
\(L(\frac12)\) \(\approx\) \(0.9097681147\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.18T + 5T^{2} \)
7 \( 1 + 1.10T + 7T^{2} \)
17 \( 1 + 6.29T + 17T^{2} \)
19 \( 1 + 3.74T + 19T^{2} \)
23 \( 1 + 3.26T + 23T^{2} \)
29 \( 1 - 3.60T + 29T^{2} \)
31 \( 1 - 0.336T + 31T^{2} \)
37 \( 1 - 6.37T + 37T^{2} \)
41 \( 1 - 0.894T + 41T^{2} \)
43 \( 1 + 8.85T + 43T^{2} \)
47 \( 1 - 8.21T + 47T^{2} \)
53 \( 1 - 5.32T + 53T^{2} \)
59 \( 1 - 9.01T + 59T^{2} \)
61 \( 1 + 14.9T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 - 5.32T + 71T^{2} \)
73 \( 1 + 5.03T + 73T^{2} \)
79 \( 1 + 9.54T + 79T^{2} \)
83 \( 1 - 5.11T + 83T^{2} \)
89 \( 1 - 3.66T + 89T^{2} \)
97 \( 1 - 8.13T + 97T^{2} \)
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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

−7.901117302951875777942755990484, −7.44871357579514465479321819018, −6.67230183754304370464556390726, −6.06432031103699448321016858177, −4.72023130585570917220570456472, −4.36320732904890544979701464020, −3.61572676144297510528591599137, −2.81200120341979243166919960193, −2.00844397431474141938390203981, −0.44786720538132208449908148316, 0.44786720538132208449908148316, 2.00844397431474141938390203981, 2.81200120341979243166919960193, 3.61572676144297510528591599137, 4.36320732904890544979701464020, 4.72023130585570917220570456472, 6.06432031103699448321016858177, 6.67230183754304370464556390726, 7.44871357579514465479321819018, 7.901117302951875777942755990484

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