Properties

Label 2-1386-1.1-c3-0-71
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 + 7·5-s + 7·7-s + 8·8-s + 14·10-s − 11·11-s − 67·13-s + 14·14-s + 16·16-s − 30·17-s − 7·19-s + 28·20-s − 22·22-s − 28·23-s − 76·25-s − 134·26-s + 28·28-s − 121·29-s − 310·31-s + 32·32-s − 60·34-s + 49·35-s − 71·37-s − 14·38-s + 56·40-s + 180·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.626·5-s + 0.377·7-s + 0.353·8-s + 0.442·10-s − 0.301·11-s − 1.42·13-s + 0.267·14-s + 1/4·16-s − 0.428·17-s − 0.0845·19-s + 0.313·20-s − 0.213·22-s − 0.253·23-s − 0.607·25-s − 1.01·26-s + 0.188·28-s − 0.774·29-s − 1.79·31-s + 0.176·32-s − 0.302·34-s + 0.236·35-s − 0.315·37-s − 0.0597·38-s + 0.221·40-s + 0.685·41-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 \( 1 - 7 T + p^{3} T^{2} \)
13 \( 1 + 67 T + p^{3} T^{2} \)
17 \( 1 + 30 T + p^{3} T^{2} \)
19 \( 1 + 7 T + p^{3} T^{2} \)
23 \( 1 + 28 T + p^{3} T^{2} \)
29 \( 1 + 121 T + p^{3} T^{2} \)
31 \( 1 + 10 p T + p^{3} T^{2} \)
37 \( 1 + 71 T + p^{3} T^{2} \)
41 \( 1 - 180 T + p^{3} T^{2} \)
43 \( 1 + 108 T + p^{3} T^{2} \)
47 \( 1 + 71 T + p^{3} T^{2} \)
53 \( 1 + 128 T + p^{3} T^{2} \)
59 \( 1 - 429 T + p^{3} T^{2} \)
61 \( 1 - 22 T + p^{3} T^{2} \)
67 \( 1 + 803 T + p^{3} T^{2} \)
71 \( 1 + 468 T + p^{3} T^{2} \)
73 \( 1 + 117 T + p^{3} T^{2} \)
79 \( 1 + 96 T + p^{3} T^{2} \)
83 \( 1 - 1122 T + p^{3} T^{2} \)
89 \( 1 - 1146 T + p^{3} T^{2} \)
97 \( 1 + 92 T + p^{3} T^{2} \)
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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

−8.904838613736194506378769968733, −7.70061384693885111434857815193, −7.21726781719692113450559084611, −6.12271885235008361613748041409, −5.38216107645771569084856371447, −4.69457782279696940628856385650, −3.65563577989240869784434108319, −2.43489982006772064109432302748, −1.78459722927078568737247054476, 0, 1.78459722927078568737247054476, 2.43489982006772064109432302748, 3.65563577989240869784434108319, 4.69457782279696940628856385650, 5.38216107645771569084856371447, 6.12271885235008361613748041409, 7.21726781719692113450559084611, 7.70061384693885111434857815193, 8.904838613736194506378769968733

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