Properties

Label 2-6384-1.1-c1-0-94
Degree $2$
Conductor $6384$
Sign $-1$
Analytic cond. $50.9764$
Root an. cond. $7.13978$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 7-s + 9-s + 2·13-s − 2·15-s − 2·17-s − 19-s − 21-s − 4·23-s − 25-s − 27-s − 2·29-s + 2·35-s − 2·37-s − 2·39-s − 6·41-s − 4·43-s + 2·45-s + 49-s + 2·51-s − 10·53-s + 57-s − 12·59-s − 10·61-s + 63-s + 4·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 0.485·17-s − 0.229·19-s − 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.338·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1/7·49-s + 0.280·51-s − 1.37·53-s + 0.132·57-s − 1.56·59-s − 1.28·61-s + 0.125·63-s + 0.496·65-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: yes
Arithmetic: yes
Character: $\chi_{6384} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6384,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.71598464208158780765361859731, −6.73456560568447262229305324722, −6.22963690808262854308155816900, −5.62557596616354097945717516585, −4.89870089264221382961397004508, −4.15948364763880779739713392620, −3.20381309730681673396081067438, −2.02653962005260481268813451036, −1.48127017826735144571644996023, 0, 1.48127017826735144571644996023, 2.02653962005260481268813451036, 3.20381309730681673396081067438, 4.15948364763880779739713392620, 4.89870089264221382961397004508, 5.62557596616354097945717516585, 6.22963690808262854308155816900, 6.73456560568447262229305324722, 7.71598464208158780765361859731

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