Properties

Label 2-6384-1.1-c1-0-81
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 + 6·17-s + 19-s − 21-s − 25-s + 27-s − 10·29-s + 4·31-s + 2·35-s + 2·37-s − 2·39-s + 10·41-s − 4·43-s − 2·45-s − 12·47-s + 49-s + 6·51-s − 2·53-s + 57-s + 4·59-s − 2·61-s − 63-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 + 1.45·17-s + 0.229·19-s − 0.218·21-s − 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.338·35-s + 0.328·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s − 0.298·45-s − 1.75·47-s + 1/7·49-s + 0.840·51-s − 0.274·53-s + 0.132·57-s + 0.520·59-s − 0.256·61-s − 0.125·63-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: Trivial
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 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.74106590416208949471647164739, −7.26421137267251213890256578118, −6.31470470956331558749608234664, −5.51604012635626845778795752343, −4.68519738993808231019979345348, −3.80209096539206879023991289283, −3.33855060444617927870814231532, −2.46353436514245068817483246561, −1.30165177294411114716945105693, 0, 1.30165177294411114716945105693, 2.46353436514245068817483246561, 3.33855060444617927870814231532, 3.80209096539206879023991289283, 4.68519738993808231019979345348, 5.51604012635626845778795752343, 6.31470470956331558749608234664, 7.26421137267251213890256578118, 7.74106590416208949471647164739

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