Properties

Label 2-6384-1.1-c1-0-10
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 − 3.88·5-s − 7-s + 9-s − 3.21·11-s + 5.39·13-s − 3.88·15-s − 4.41·17-s − 19-s − 21-s + 3.39·23-s + 10.1·25-s + 27-s − 7.28·29-s − 0.187·31-s − 3.21·33-s + 3.88·35-s − 4.60·37-s + 5.39·39-s − 9.77·41-s − 3.88·45-s − 4.45·47-s + 49-s − 4.41·51-s + 11.2·53-s + 12.4·55-s − 57-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.73·5-s − 0.377·7-s + 0.333·9-s − 0.968·11-s + 1.49·13-s − 1.00·15-s − 1.06·17-s − 0.229·19-s − 0.218·21-s + 0.708·23-s + 2.02·25-s + 0.192·27-s − 1.35·29-s − 0.0336·31-s − 0.558·33-s + 0.657·35-s − 0.757·37-s + 0.864·39-s − 1.52·41-s − 0.579·45-s − 0.649·47-s + 0.142·49-s − 0.617·51-s + 1.55·53-s + 1.68·55-s − 0.132·57-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

\(L(1)\) \(\approx\) \(1.082877918\)
\(L(\frac12)\) \(\approx\) \(1.082877918\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.88T + 5T^{2} \)
11 \( 1 + 3.21T + 11T^{2} \)
13 \( 1 - 5.39T + 13T^{2} \)
17 \( 1 + 4.41T + 17T^{2} \)
23 \( 1 - 3.39T + 23T^{2} \)
29 \( 1 + 7.28T + 29T^{2} \)
31 \( 1 + 0.187T + 31T^{2} \)
37 \( 1 + 4.60T + 37T^{2} \)
41 \( 1 + 9.77T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 4.45T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 - 5.77T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 2.48T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 4.08T + 73T^{2} \)
79 \( 1 + 2.68T + 79T^{2} \)
83 \( 1 + 3.92T + 83T^{2} \)
89 \( 1 - 2.70T + 89T^{2} \)
97 \( 1 + 0.227T + 97T^{2} \)
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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

−8.180147352473643151421308468050, −7.24861754501999214291783797990, −7.00114881667931914955245309629, −5.95438544259550959605505953437, −4.98297152109937953216006489351, −4.19610252724115562262987292962, −3.56281979919637936154614597714, −3.07188853223184423391886889393, −1.90036113078400727492739229611, −0.50909016503966996239391844152, 0.50909016503966996239391844152, 1.90036113078400727492739229611, 3.07188853223184423391886889393, 3.56281979919637936154614597714, 4.19610252724115562262987292962, 4.98297152109937953216006489351, 5.95438544259550959605505953437, 7.00114881667931914955245309629, 7.24861754501999214291783797990, 8.180147352473643151421308468050

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