Properties

Label 2-6384-1.1-c1-0-11
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 − 3.33·5-s + 7-s + 9-s − 4.27·11-s − 3.33·15-s − 6.05·17-s + 19-s + 21-s − 3.25·23-s + 6.12·25-s + 27-s + 8.45·29-s − 2.27·31-s − 4.27·33-s − 3.33·35-s − 0.724·37-s − 0.127·41-s + 8.99·43-s − 3.33·45-s − 7.47·47-s + 49-s − 6.05·51-s + 2.21·53-s + 14.2·55-s + 57-s − 4.79·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.49·5-s + 0.377·7-s + 0.333·9-s − 1.28·11-s − 0.861·15-s − 1.46·17-s + 0.229·19-s + 0.218·21-s − 0.677·23-s + 1.22·25-s + 0.192·27-s + 1.57·29-s − 0.407·31-s − 0.743·33-s − 0.563·35-s − 0.119·37-s − 0.0198·41-s + 1.37·43-s − 0.497·45-s − 1.09·47-s + 0.142·49-s − 0.848·51-s + 0.303·53-s + 1.92·55-s + 0.132·57-s − 0.624·59-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: $\chi_{6384} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6384,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.188462782\)
\(L(\frac12)\) \(\approx\) \(1.188462782\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.33T + 5T^{2} \)
11 \( 1 + 4.27T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 6.05T + 17T^{2} \)
23 \( 1 + 3.25T + 23T^{2} \)
29 \( 1 - 8.45T + 29T^{2} \)
31 \( 1 + 2.27T + 31T^{2} \)
37 \( 1 + 0.724T + 37T^{2} \)
41 \( 1 + 0.127T + 41T^{2} \)
43 \( 1 - 8.99T + 43T^{2} \)
47 \( 1 + 7.47T + 47T^{2} \)
53 \( 1 - 2.21T + 53T^{2} \)
59 \( 1 + 4.79T + 59T^{2} \)
61 \( 1 + 6.54T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 - 1.91T + 71T^{2} \)
73 \( 1 + 0.127T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 - 6.08T + 83T^{2} \)
89 \( 1 - 1.57T + 89T^{2} \)
97 \( 1 + 4.39T + 97T^{2} \)
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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

−7.966862553537395311815723947394, −7.57644480010168831202537444242, −6.86018901138345790249066965653, −5.95015230388809409057853846168, −4.72854082392121081048973230455, −4.55077617903629729282761445609, −3.58416505730526529490629768261, −2.83913192923825981371787810362, −2.01103119428951373595939811269, −0.52616893575508350564251437079, 0.52616893575508350564251437079, 2.01103119428951373595939811269, 2.83913192923825981371787810362, 3.58416505730526529490629768261, 4.55077617903629729282761445609, 4.72854082392121081048973230455, 5.95015230388809409057853846168, 6.86018901138345790249066965653, 7.57644480010168831202537444242, 7.966862553537395311815723947394

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