Properties

Label 2-1386-1.1-c3-0-64
Degree $2$
Conductor $1386$
Sign $-1$
Analytic cond. $81.7766$
Root an. cond. $9.04304$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 14·5-s + 7·7-s − 8·8-s − 28·10-s + 11·11-s − 16·13-s − 14·14-s + 16·16-s − 108·17-s + 116·19-s + 56·20-s − 22·22-s − 68·23-s + 71·25-s + 32·26-s + 28·28-s − 122·29-s − 262·31-s − 32·32-s + 216·34-s + 98·35-s + 130·37-s − 232·38-s − 112·40-s − 204·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.25·5-s + 0.377·7-s − 0.353·8-s − 0.885·10-s + 0.301·11-s − 0.341·13-s − 0.267·14-s + 1/4·16-s − 1.54·17-s + 1.40·19-s + 0.626·20-s − 0.213·22-s − 0.616·23-s + 0.567·25-s + 0.241·26-s + 0.188·28-s − 0.781·29-s − 1.51·31-s − 0.176·32-s + 1.08·34-s + 0.473·35-s + 0.577·37-s − 0.990·38-s − 0.442·40-s − 0.777·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(81.7766\)
Root analytic conductor: \(9.04304\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1386} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1386,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 \)
7 \( 1 - p T \)
11 \( 1 - p T \)
good5 \( 1 - 14 T + p^{3} T^{2} \)
13 \( 1 + 16 T + p^{3} T^{2} \)
17 \( 1 + 108 T + p^{3} T^{2} \)
19 \( 1 - 116 T + p^{3} T^{2} \)
23 \( 1 + 68 T + p^{3} T^{2} \)
29 \( 1 + 122 T + p^{3} T^{2} \)
31 \( 1 + 262 T + p^{3} T^{2} \)
37 \( 1 - 130 T + p^{3} T^{2} \)
41 \( 1 + 204 T + p^{3} T^{2} \)
43 \( 1 + 396 T + p^{3} T^{2} \)
47 \( 1 + 166 T + p^{3} T^{2} \)
53 \( 1 + 442 T + p^{3} T^{2} \)
59 \( 1 + 702 T + p^{3} T^{2} \)
61 \( 1 - 196 T + p^{3} T^{2} \)
67 \( 1 + 416 T + p^{3} T^{2} \)
71 \( 1 + 492 T + p^{3} T^{2} \)
73 \( 1 - 408 T + p^{3} T^{2} \)
79 \( 1 - 600 T + p^{3} T^{2} \)
83 \( 1 - 1212 T + p^{3} T^{2} \)
89 \( 1 + 1146 T + p^{3} T^{2} \)
97 \( 1 + 482 T + p^{3} T^{2} \)
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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

−9.102070693262266012995891336730, −8.062882904890862625651410012621, −7.20634538434498624830327614029, −6.40389452810723188627414594486, −5.60687993451256688531131076496, −4.73811548226100006340100167694, −3.34855889017840624999712561016, −2.10789422559820437720030141443, −1.55296442445375211958552898971, 0, 1.55296442445375211958552898971, 2.10789422559820437720030141443, 3.34855889017840624999712561016, 4.73811548226100006340100167694, 5.60687993451256688531131076496, 6.40389452810723188627414594486, 7.20634538434498624830327614029, 8.062882904890862625651410012621, 9.102070693262266012995891336730

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