Properties

Label 2-6384-1.1-c1-0-49
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 + 1.69·5-s + 7-s + 9-s − 1.32·11-s + 1.69·15-s + 0.102·17-s + 19-s + 21-s + 9.16·23-s − 2.12·25-s + 27-s − 4.81·29-s + 0.675·31-s − 1.32·33-s + 1.69·35-s + 0.406·37-s + 4.04·41-s + 4.91·43-s + 1.69·45-s − 3.66·47-s + 49-s + 0.102·51-s + 5.42·53-s − 2.24·55-s + 57-s + 9.43·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.758·5-s + 0.377·7-s + 0.333·9-s − 0.399·11-s + 0.437·15-s + 0.0248·17-s + 0.229·19-s + 0.218·21-s + 1.91·23-s − 0.424·25-s + 0.192·27-s − 0.894·29-s + 0.121·31-s − 0.230·33-s + 0.286·35-s + 0.0668·37-s + 0.631·41-s + 0.749·43-s + 0.252·45-s − 0.535·47-s + 0.142·49-s + 0.0143·51-s + 0.745·53-s − 0.302·55-s + 0.132·57-s + 1.22·59-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

\(L(1)\) \(\approx\) \(3.230078876\)
\(L(\frac12)\) \(\approx\) \(3.230078876\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.69T + 5T^{2} \)
11 \( 1 + 1.32T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 0.102T + 17T^{2} \)
23 \( 1 - 9.16T + 23T^{2} \)
29 \( 1 + 4.81T + 29T^{2} \)
31 \( 1 - 0.675T + 31T^{2} \)
37 \( 1 - 0.406T + 37T^{2} \)
41 \( 1 - 4.04T + 41T^{2} \)
43 \( 1 - 4.91T + 43T^{2} \)
47 \( 1 + 3.66T + 47T^{2} \)
53 \( 1 - 5.42T + 53T^{2} \)
59 \( 1 - 9.43T + 59T^{2} \)
61 \( 1 + 0.648T + 61T^{2} \)
67 \( 1 - 8.92T + 67T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 - 4.04T + 73T^{2} \)
79 \( 1 + 8.55T + 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 + 4.85T + 89T^{2} \)
97 \( 1 - 2.71T + 97T^{2} \)
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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

−8.036356384867615772652955748174, −7.34988166866475462376034545267, −6.75816660479971114002264731419, −5.75937375457307341100970097884, −5.25963112587820928832869960347, −4.43009943587256971514688668947, −3.50856275869570646329338710255, −2.66139993583254413027002775825, −1.95914614885260141209475623186, −0.938960146235187582717644318466, 0.938960146235187582717644318466, 1.95914614885260141209475623186, 2.66139993583254413027002775825, 3.50856275869570646329338710255, 4.43009943587256971514688668947, 5.25963112587820928832869960347, 5.75937375457307341100970097884, 6.75816660479971114002264731419, 7.34988166866475462376034545267, 8.036356384867615772652955748174

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