Properties

Label 2-6384-1.1-c1-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s − 7-s + 9-s + 2·11-s − 6·13-s + 2·15-s + 2·17-s − 19-s + 21-s − 6·23-s − 25-s − 27-s − 8·29-s + 8·31-s − 2·33-s + 2·35-s − 10·37-s + 6·39-s + 8·41-s + 8·43-s − 2·45-s − 2·47-s + 49-s − 2·51-s − 8·53-s − 4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s + 0.516·15-s + 0.485·17-s − 0.229·19-s + 0.218·21-s − 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.43·31-s − 0.348·33-s + 0.338·35-s − 1.64·37-s + 0.960·39-s + 1.24·41-s + 1.21·43-s − 0.298·45-s − 0.291·47-s + 1/7·49-s − 0.280·51-s − 1.09·53-s − 0.539·55-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: \(0\)
Selberg data: \((2,\ 6384,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5711191311\)
\(L(\frac12)\) \(\approx\) \(0.5711191311\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−7.68232433615238804170965301329, −7.51424911854230365986047208393, −6.62504009228692607135522425819, −5.93577847671402433553300941243, −5.16355566525291829630601803053, −4.31338718100233797373331504302, −3.83197538552969534374276864475, −2.80963779812988033436297944151, −1.77927837757618721073394184152, −0.39207630584210326153724945313, 0.39207630584210326153724945313, 1.77927837757618721073394184152, 2.80963779812988033436297944151, 3.83197538552969534374276864475, 4.31338718100233797373331504302, 5.16355566525291829630601803053, 5.93577847671402433553300941243, 6.62504009228692607135522425819, 7.51424911854230365986047208393, 7.68232433615238804170965301329

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