Properties

Label 2-6864-1.1-c1-0-35
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 − 3.47·5-s + 3.67·7-s + 9-s + 11-s + 13-s − 3.47·15-s + 2.76·17-s − 6.53·19-s + 3.67·21-s + 2.05·23-s + 7.06·25-s + 27-s + 5.71·29-s − 4.73·31-s + 33-s − 12.7·35-s − 5.04·37-s + 39-s + 9.96·41-s − 0.157·43-s − 3.47·45-s + 12.5·47-s + 6.51·49-s + 2.76·51-s − 9.05·53-s − 3.47·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.55·5-s + 1.38·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s − 0.896·15-s + 0.669·17-s − 1.49·19-s + 0.802·21-s + 0.428·23-s + 1.41·25-s + 0.192·27-s + 1.06·29-s − 0.850·31-s + 0.174·33-s − 2.15·35-s − 0.829·37-s + 0.160·39-s + 1.55·41-s − 0.0240·43-s − 0.517·45-s + 1.83·47-s + 0.930·49-s + 0.386·51-s − 1.24·53-s − 0.468·55-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

\(L(1)\) \(\approx\) \(2.263457150\)
\(L(\frac12)\) \(\approx\) \(2.263457150\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.47T + 5T^{2} \)
7 \( 1 - 3.67T + 7T^{2} \)
17 \( 1 - 2.76T + 17T^{2} \)
19 \( 1 + 6.53T + 19T^{2} \)
23 \( 1 - 2.05T + 23T^{2} \)
29 \( 1 - 5.71T + 29T^{2} \)
31 \( 1 + 4.73T + 31T^{2} \)
37 \( 1 + 5.04T + 37T^{2} \)
41 \( 1 - 9.96T + 41T^{2} \)
43 \( 1 + 0.157T + 43T^{2} \)
47 \( 1 - 12.5T + 47T^{2} \)
53 \( 1 + 9.05T + 53T^{2} \)
59 \( 1 + 0.0906T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 - 0.535T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 6.77T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + 7.25T + 89T^{2} \)
97 \( 1 + 0.269T + 97T^{2} \)
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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

−7.961742583815653114865735506731, −7.48322942847417585585147441332, −6.86079446905771971678130673411, −5.80863931873110261911661503387, −4.83429295876995722151531220670, −4.25351246549694089148415270729, −3.77186397945876276305305303824, −2.80851242519473359949302161853, −1.79261957530243163789188962942, −0.77377217360794063005062725911, 0.77377217360794063005062725911, 1.79261957530243163789188962942, 2.80851242519473359949302161853, 3.77186397945876276305305303824, 4.25351246549694089148415270729, 4.83429295876995722151531220670, 5.80863931873110261911661503387, 6.86079446905771971678130673411, 7.48322942847417585585147441332, 7.961742583815653114865735506731

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