Properties

Label 2-6384-1.1-c1-0-6
Degree $2$
Conductor $6384$
Sign $1$
Analytic cond. $50.9764$
Root an. cond. $7.13978$
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 − 2.73·5-s + 7-s + 9-s − 3.46·11-s − 4·13-s + 2.73·15-s + 4.19·17-s + 19-s − 21-s + 7.46·23-s + 2.46·25-s − 27-s − 6.19·29-s − 4.92·31-s + 3.46·33-s − 2.73·35-s − 7.46·37-s + 4·39-s − 9.46·41-s + 3.46·43-s − 2.73·45-s − 2.73·47-s + 49-s − 4.19·51-s + 8.73·53-s + 9.46·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.22·5-s + 0.377·7-s + 0.333·9-s − 1.04·11-s − 1.10·13-s + 0.705·15-s + 1.01·17-s + 0.229·19-s − 0.218·21-s + 1.55·23-s + 0.492·25-s − 0.192·27-s − 1.15·29-s − 0.885·31-s + 0.603·33-s − 0.461·35-s − 1.22·37-s + 0.640·39-s − 1.47·41-s + 0.528·43-s − 0.407·45-s − 0.398·47-s + 0.142·49-s − 0.587·51-s + 1.19·53-s + 1.27·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6384\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(50.9764\)
Root analytic conductor: \(7.13978\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6384,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6526112130\)
\(L(\frac12)\) \(\approx\) \(0.6526112130\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 - T \)
19 \( 1 - T \)
good5 \( 1 + 2.73T + 5T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 4.19T + 17T^{2} \)
23 \( 1 - 7.46T + 23T^{2} \)
29 \( 1 + 6.19T + 29T^{2} \)
31 \( 1 + 4.92T + 31T^{2} \)
37 \( 1 + 7.46T + 37T^{2} \)
41 \( 1 + 9.46T + 41T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 + 2.73T + 47T^{2} \)
53 \( 1 - 8.73T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 + 1.46T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + 15.8T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 1.26T + 83T^{2} \)
89 \( 1 - 9.46T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−7.75376337644143467283408474966, −7.40180112364703885069882109622, −6.93199094471312175867879323915, −5.57223473982031233498518546081, −5.23465130331779043845031751241, −4.55459432795056270482395860467, −3.60802639950858670481054499642, −2.93182901094852448434233079614, −1.71955712537866695108717160281, −0.42601561056753706415540454115, 0.42601561056753706415540454115, 1.71955712537866695108717160281, 2.93182901094852448434233079614, 3.60802639950858670481054499642, 4.55459432795056270482395860467, 5.23465130331779043845031751241, 5.57223473982031233498518546081, 6.93199094471312175867879323915, 7.40180112364703885069882109622, 7.75376337644143467283408474966

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