Properties

Label 2-6384-1.1-c1-0-56
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 + 0.841·5-s − 7-s + 9-s + 4.29·11-s + 5.02·13-s + 0.841·15-s + 6.86·17-s − 19-s − 21-s + 3.02·23-s − 4.29·25-s + 27-s − 2.18·29-s − 7.31·31-s + 4.29·33-s − 0.841·35-s + 3.27·37-s + 5.02·39-s − 0.317·41-s + 0.841·45-s + 12.7·47-s + 49-s + 6.86·51-s + 6.18·53-s + 3.61·55-s − 57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.376·5-s − 0.377·7-s + 0.333·9-s + 1.29·11-s + 1.39·13-s + 0.217·15-s + 1.66·17-s − 0.229·19-s − 0.218·21-s + 0.630·23-s − 0.858·25-s + 0.192·27-s − 0.404·29-s − 1.31·31-s + 0.747·33-s − 0.142·35-s + 0.537·37-s + 0.804·39-s − 0.0495·41-s + 0.125·45-s + 1.85·47-s + 0.142·49-s + 0.960·51-s + 0.849·53-s + 0.487·55-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

\(L(1)\) \(\approx\) \(3.365905139\)
\(L(\frac12)\) \(\approx\) \(3.365905139\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.841T + 5T^{2} \)
11 \( 1 - 4.29T + 11T^{2} \)
13 \( 1 - 5.02T + 13T^{2} \)
17 \( 1 - 6.86T + 17T^{2} \)
23 \( 1 - 3.02T + 23T^{2} \)
29 \( 1 + 2.18T + 29T^{2} \)
31 \( 1 + 7.31T + 31T^{2} \)
37 \( 1 - 3.27T + 37T^{2} \)
41 \( 1 + 0.317T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 - 6.18T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 1.86T + 67T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 + 2.74T + 73T^{2} \)
79 \( 1 + 1.72T + 79T^{2} \)
83 \( 1 - 6.72T + 83T^{2} \)
89 \( 1 - 3.29T + 89T^{2} \)
97 \( 1 + 1.43T + 97T^{2} \)
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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

−8.012036632556918100389508077960, −7.38987550778230063329314216334, −6.58309030726029555090697081718, −5.93024774509983341320232385770, −5.37236266987759622651432515796, −3.94700501772623750320519095977, −3.79331950864390934321463922751, −2.86493014811589384672205362476, −1.71508622439115764832889996288, −1.02412686354337809843572197497, 1.02412686354337809843572197497, 1.71508622439115764832889996288, 2.86493014811589384672205362476, 3.79331950864390934321463922751, 3.94700501772623750320519095977, 5.37236266987759622651432515796, 5.93024774509983341320232385770, 6.58309030726029555090697081718, 7.38987550778230063329314216334, 8.012036632556918100389508077960

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