Properties

Label 2-6864-1.1-c1-0-55
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 + 0.169·5-s + 1.46·7-s + 9-s + 11-s + 13-s + 0.169·15-s + 6.72·17-s + 6.80·19-s + 1.46·21-s − 5.29·23-s − 4.97·25-s + 27-s − 4.23·29-s + 7.17·31-s + 33-s + 0.247·35-s + 5.03·37-s + 39-s + 0.720·41-s − 12.1·43-s + 0.169·45-s + 10.4·47-s − 4.86·49-s + 6.72·51-s + 12.9·53-s + 0.169·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.0757·5-s + 0.552·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 0.0437·15-s + 1.63·17-s + 1.56·19-s + 0.319·21-s − 1.10·23-s − 0.994·25-s + 0.192·27-s − 0.785·29-s + 1.28·31-s + 0.174·33-s + 0.0418·35-s + 0.826·37-s + 0.160·39-s + 0.112·41-s − 1.85·43-s + 0.0252·45-s + 1.52·47-s − 0.694·49-s + 0.941·51-s + 1.77·53-s + 0.0228·55-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

\(L(1)\) \(\approx\) \(3.259538123\)
\(L(\frac12)\) \(\approx\) \(3.259538123\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.169T + 5T^{2} \)
7 \( 1 - 1.46T + 7T^{2} \)
17 \( 1 - 6.72T + 17T^{2} \)
19 \( 1 - 6.80T + 19T^{2} \)
23 \( 1 + 5.29T + 23T^{2} \)
29 \( 1 + 4.23T + 29T^{2} \)
31 \( 1 - 7.17T + 31T^{2} \)
37 \( 1 - 5.03T + 37T^{2} \)
41 \( 1 - 0.720T + 41T^{2} \)
43 \( 1 + 12.1T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 + 3.92T + 59T^{2} \)
61 \( 1 - 1.23T + 61T^{2} \)
67 \( 1 + 0.504T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 3.12T + 73T^{2} \)
79 \( 1 + 6.05T + 79T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 + 1.80T + 89T^{2} \)
97 \( 1 + 9.83T + 97T^{2} \)
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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

−7.916418455636406128949142250716, −7.54967524772402683349835054316, −6.62214337995571700474112645972, −5.70392716650886999355986641355, −5.27351816880945719861240226081, −4.17959298896063715097702653429, −3.60155283944887728569939878866, −2.78610223433225189142919449863, −1.76006028555787896386126410809, −0.968665162519282270619096063035, 0.968665162519282270619096063035, 1.76006028555787896386126410809, 2.78610223433225189142919449863, 3.60155283944887728569939878866, 4.17959298896063715097702653429, 5.27351816880945719861240226081, 5.70392716650886999355986641355, 6.62214337995571700474112645972, 7.54967524772402683349835054316, 7.916418455636406128949142250716

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